Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9
ANSWERS FOR WORKBOOKS The answers for Chapter 10 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
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ISBN: 0-07-828013-3 Algebra 2Chapter 10 Resource Masters
1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02
© Glencoe/McGraw-Hill iii Glencoe Algebra 2
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 10-1Study Guide and Intervention . . . . . . . . 573–574Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 575Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 576Reading to Learn Mathematics . . . . . . . . . . 577Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 578
Lesson 10-2Study Guide and Intervention . . . . . . . . 579–580Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 581Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 582Reading to Learn Mathematics . . . . . . . . . . 583Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 584
Lesson 10-3Study Guide and Intervention . . . . . . . . 585–586Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 587Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 588Reading to Learn Mathematics . . . . . . . . . . 589Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 590
Lesson 10-4Study Guide and Intervention . . . . . . . . 591–592Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 593Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 594Reading to Learn Mathematics . . . . . . . . . . 595Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 596
Lesson 10-5Study Guide and Intervention . . . . . . . . 597–598Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 599Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 600Reading to Learn Mathematics . . . . . . . . . . 601Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 602
Lesson 10-6Study Guide and Intervention . . . . . . . . 603–604Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 605Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 606Reading to Learn Mathematics . . . . . . . . . . 607Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 608
Chapter 10 AssessmentChapter 10 Test, Form 1 . . . . . . . . . . . 609–610Chapter 10 Test, Form 2A . . . . . . . . . . 611–612Chapter 10 Test, Form 2B . . . . . . . . . . 613–614Chapter 10 Test, Form 2C . . . . . . . . . . 615–616Chapter 10 Test, Form 2D . . . . . . . . . . 617–618Chapter 10 Test, Form 3 . . . . . . . . . . . 619–620Chapter 10 Open-Ended Assessment . . . . . 621Chapter 10 Vocabulary Test/Review . . . . . . 622Chapter 10 Quizzes 1 & 2 . . . . . . . . . . . . . . 623Chapter 10 Quizzes 3 & 4 . . . . . . . . . . . . . . 624Chapter 10 Mid-Chapter Test . . . . . . . . . . . . 625Chapter 10 Cumulative Review . . . . . . . . . . 626Chapter 10 Standardized Test Practice . 627–628Unit 3 Test/Review (Ch. 8–10) . . . . . . . 629–630
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A30
© Glencoe/McGraw-Hill iv Glencoe Algebra 2
Teacher’s Guide to Using theChapter 10 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 10 Resource Masters includes the core materialsneeded for Chapter 10. These materials include worksheets, extensions, andassessment options. The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 10-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.
Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
© Glencoe/McGraw-Hill v Glencoe Algebra 2
Assessment OptionsThe assessment masters in the Chapter 10Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 572–573. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
1010
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 10.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
Change of Base Formula
common logarithm
LAW·guh·RIH·thuhm
exponential decay
EHK·spuh·NEHN·chuhl
exponential equation
exponential function
exponential growth
exponential inequality
(continued on the next page)
© Glencoe/McGraw-Hill vii Glencoe Algebra 2
© Glencoe/McGraw-Hill viii Glencoe Algebra 2
Vocabulary Term Found on Page Definition/Description/Example
logarithm
logarithmic function
LAW·guh·RIHTH·mihk
natural base, e
natural base exponential function
natural logarithm
natural logarithmic function
rate of decay
rate of growth
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
1010
Study Guide and InterventionExponential Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
© Glencoe/McGraw-Hill 573 Glencoe Algebra 2
Less
on
10-
1
Exponential Functions An exponential function has the form y ! abx,where a " 0, b # 0, and b " 1.
1. The function is continuous and one-to-one.
Properties of an2. The domain is the set of all real numbers.
Exponential Function3. The x-axis is the asymptote of the graph.4. The range is the set of all positive numbers if a # 0 and all negative numbers if a $ 0.5. The graph contains the point (0, a).
Exponential Growth If a # 0 and b # 1, the function y ! abx represents exponential growth.and Decay If a # 0 and 0 $ b $ 1, the function y ! abx represents exponential decay.
Sketch the graph of y ! 0.1(4)x. Then state the function’s domain and range.Make a table of values. Connect the points to form a smooth curve.
The domain of the function is all real numbers, while the range is the set of all positive real numbers.
Determine whether each function represents exponential growth or decay.a. y ! 0.5(2)x b. y ! %2.8(2)x c. y ! 1.1(0.5)x
exponential growth, neither, since %2.8, exponential decay, sincesince the base, 2, is the value of a is less the base, 0.5, is betweengreater than 1 than 0. 0 and 1
Sketch the graph of each function. Then state the function’s domain and range.
1. y ! 3(2)x 2. y ! %2! "x3. y ! 0.25(5)x
Domain: all real Domain: all real Domain: all real numbers; Range: all numbers; Range: all numbers; Range: allpositive real numbers negative real numbers positive real numbers
Determine whether each function represents exponential growth or decay.
4. y ! 0.3(1.2)x growth 5. y ! %5! "xneither 6. y ! 3(10)%x decay4
&5
x
y
O
x
y
O
x
y
O
1&4
x %1 0 1 2 3
y 0.025 0.1 0.4 1.6 6.4
x
y
O
Example 1Example 1
Example 2Example 2
ExercisesExercises
© Glencoe/McGraw-Hill 574 Glencoe Algebra 2
Exponential Equations and Inequalities All the properties of rational exponentsthat you know also apply to real exponents. Remember that am ' an ! am ( n, (am)n ! amn,and am ) an ! am % n.
Property of Equality for If b is a positive number other than 1,Exponential Functions then bx ! by if and only if x ! y.
Property of Inequality forIf b # 1
Exponential Functionsthen bx # by if and only if x # yand bx $ by if and only if x $ y.
Study Guide and Intervention (continued)
Exponential Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
Solve 4x " 1 ! 2x # 5.4x % 1 ! 2x ( 5 Original equation
(22)x % 1 ! 2x ( 5 Rewrite 4 as 22.
2(x % 1) ! x ( 5 Prop. of Inequality for ExponentialFunctions
2x % 2 ! x ( 5 Distributive Property
x ! 7 Subtract x and add 2 to each side.
Solve 52x " 1 $ .
52x % 1 # Original inequality
52x % 1 # 5%3 Rewrite as 5%3.
2x % 1 # %3 Prop. of Inequality for Exponential Functions
2x # %2 Add 1 to each side.
x # %1 Divide each side by 2.
The solution set is {x|x # %1}.
1&125
1&125
1%125
Example 1Example 1 Example 2Example 2
ExercisesExercises
Simplify each expression.
1. (3#2$)#2$ 2. 25#2$ ' 125#2$ 3. (x#2$y3#2$)#2$
9 55!2" or 3125!2" x2y6
4. (x#6$)(x#5$) 5. (x#6$)#5$ 6. (2x*)(5x3*)x!6" # !5" x!30" 10x4&
Solve each equation or inequality. Check your solution.
7. 32x % 1 ! 3x ( 2 3 8. 23x ! 4x ( 2 4 9. 32x % 1 ! "
10. 4x ( 1 ! 82x ( 3 " 11. 8x % 2 ! 12. 252x ! 125x ( 2 6
13. 4#x$ ! 16#5$ 20 14. x#3$ ! 36%&&34& 6 15. x#2$ ! 81$
&3
16. 3x % 4 $ x ' 1 17. 42x % 2 # 2x ( 1 x $ 18. 52x $ 125x % 5 x $ 15
19. 104x ( 1 # 100x % 2 20. 73x $ 49x2 21. 82x % 5 $ 4x ( 8
x $ " x $ or x ' 0 x ' %341%
3%2
5%2
5%3
1&27
2%3
1&16
7%4
1%2
1&9
Skills PracticeExponential Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
© Glencoe/McGraw-Hill 575 Glencoe Algebra 2
Less
on
10-
1
Sketch the graph of each function. Then state the function’s domain and range.
1. y ! 3(2)x 2. y ! 2! "x
domain: all real numbers; domain: all real numbers;range: all positive numbers range: all positive numbers
Determine whether each function represents exponential growth or decay.
3. y ! 3(6)x growth 4. y ! 2! "xdecay
5. y ! 10%x decay 6. y ! 2(2.5)x growth
Write an exponential function whose graph passes through the given points.
7. (0, 1) and (%1, 3) y ! # $x8. (0, 4) and (1, 12) y ! 4(3)x
9. (0, 3) and (%1, 6) y ! 3# $x10. (0, 5) and (1, 15) y ! 5(3)x
11. (0, 0.1) and (1, 0.5) y ! 0.1(5)x 12. (0, 0.2) and (1, 1.6) y ! 0.2(8)x
Simplify each expression.
13. (3#3$)#3$ 27 14. (x#2$)#7$ x!14"
15. 52#3$ ' 54#3$ 56!3" 16. x3* ) x* x2&
Solve each equation or inequality. Check your solution.
17. 3x # 9 x $ 2 18. 22x ( 3 ! 32 1
19. 49x + x ( " 20. 43x % 2 ! 16
21. 32x ( 5 ! 27x 5 22. 27x ! 32x ( 3 3
4%3
1%2
1&7
1%2
1%3
9&10
x
y
Ox
y
O
1&2
© Glencoe/McGraw-Hill 576 Glencoe Algebra 2
Sketch the graph of each function. Then state the function’s domain and range.
1. y ! 1.5(2)x 2. y ! 4(3)x 3. y ! 3(0.5)x
domain: all real domain: all real domain: all real numbers; range: all numbers; range: all numbers; range: all positive numbers positive numbers positive numbers
Determine whether each function represents exponential growth or decay.
4. y ! 5(0.6)x decay 5. y ! 0.1(2)x growth 6. y ! 5 ' 4%x decay
Write an exponential function whose graph passes through the given points.
7. (0, 1) and (%1, 4) 8. (0, 2) and (1, 10) 9. (0, %3) and (1, %1.5)
y ! # $xy ! 2(5)x y ! "3(0.5)x
10. (0, 0.8) and (1, 1.6) 11. (0, %0.4) and (2, %10) 12. (0, *) and (3, 8*)
y ! 0.8(2)x y ! "0.4(5)x y ! &(2)x
Simplify each expression.
13. (2#2$)#8$ 16 14. (n#3$)#75$ n15 15. y#6$ ' y5#6$ y6!6"
16. 13#6$ ' 13#24$ 133!6" 17. n3 ) n* n3 " & 18. 125#11$ ) 5#11$ 52!11"
Solve each equation or inequality. Check your solution.
19. 33x % 5 # 81 x $ 3 20. 76x ! 72x % 20 "5 21. 36n % 5 $ 94n % 3 n $
22. 92x % 1 ! 27x ( 4 14 23. 23n % 1 , ! "nn ) 24. 164n % 1 ! 1282n ( 1
BIOLOGY For Exercises 25 and 26, use the following information.The initial number of bacteria in a culture is 12,000. The number after 3 days is 96,000.
25. Write an exponential function to model the population y of bacteria after x days.y ! 12,000(2)x
26. How many bacteria are there after 6 days? 768,00027. EDUCATION A college with a graduating class of 4000 students in the year 2002
predicts that it will have a graduating class of 4862 in 4 years. Write an exponentialfunction to model the number of students y in the graduating class t years after 2002.y ! 4000(1.05)t
11%2
1%6
1&8
1%2
1%4
x
y
Ox
y
O
Practice (Average)
Exponential Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
Reading to Learn MathematicsExponential Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
© Glencoe/McGraw-Hill 577 Glencoe Algebra 2
Less
on
10-
1
Pre-Activity How does an exponential function describe tournament play?
Read the introduction to Lesson 10-1 at the top of page 523 in your textbook.
How many rounds of play would be needed for a tournament with 100players? 7
Reading the Lesson
1. Indicate whether each of the following statements about the exponential function y ! 10x is true or false.
a. The domain is the set of all positive real numbers. false
b. The y-intercept is 1. true
c. The function is one-to-one. true
d. The y-axis is an asymptote of the graph. false
e. The range is the set of all real numbers. false
2. Determine whether each function represents exponential growth or decay.
a. y ! 0.2(3)x. growth b. y ! 3! "x. decay c. y ! 0.4(1.01)x. growth
3. Supply the reason for each step in the following solution of an exponential equation.
92x % 1 ! 27x Original equation
(32)2x % 1 ! (33)x Rewrite each side with a base of 3.32(2x % 1) ! 33x Power of a Power
2(2x % 1) ! 3x Property of Equality for Exponential Functions4x % 2 ! 3x Distributive Propertyx % 2 ! 0 Subtract 3x from each side.
x ! 2 Add 2 to each side.
Helping You Remember
4. One way to remember that polynomial functions and exponential functions are differentis to contrast the polynomial function y ! x2 and the exponential function y ! 2x. Tell atleast three ways they are different.
Sample answer: In y ! x2, the variable x is a base, but in y ! 2x, thevariable x is an exponent. The graph of y ! x2 is symmetric with respectto the y-axis, but the graph of y ! 2x is not. The graph of y ! x2 touchesthe x-axis at (0, 0), but the graph of y ! 2x has the x-axis as an asymptote.You can compute the value of y ! x2 mentally for x ! 100, but you cannotcompute the value of y ! 2x mentally for x ! 100.
2&5
© Glencoe/McGraw-Hill 578 Glencoe Algebra 2
Finding Solutions of xy ! yx
Perhaps you have noticed that if x and y are interchanged in equations suchas x ! y and xy ! 1, the resulting equation is equivalent to the originalequation. The same is true of the equation xy ! yx. However, findingsolutions of xy ! yx and drawing its graph is not a simple process.
Solve each problem. Assume that x and y are positive real numbers.
1. If a # 0, will (a, a) be a solution of xy ! yx? Justify your answer.
2. If c # 0, d # 0, and (c, d) is a solution of xy ! yx, will (d, c) also be a solution? Justify your answer.
3. Use 2 as a value for y in xy ! yx. The equation becomes x2 ! 2x.
a. Find equations for two functions, f(x) and g(x) that you could graph tofind the solutions of x2 ! 2x. Then graph the functions on a separatesheet of graph paper.
b. Use the graph you drew for part a to state two solutions for x2 ! 2x.Then use these solutions to state two solutions for xy ! yx.
4. In this exercise, a graphing calculator will be very helpful. Use the technique of Exercise 3 to complete the tables below. Then graph xy ! yx
for positive values of x and y. If there are asymptotes, show them in yourdiagram using dotted lines. Note that in the table, some values of y callfor one value of x, others call for two.
x
y
O
x y
4
4
5
5
8
8
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
x y
&12&
&34&
1
2
2
3
3
Study Guide and InterventionLogarithms and Logarithmic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
© Glencoe/McGraw-Hill 579 Glencoe Algebra 2
Less
on
10-
2
Logarithmic Functions and Expressions
Definition of Logarithm Let b and x be positive numbers, b " 1. The logarithm of x with base b is denoted with Base b logb x and is defined as the exponent y that makes the equation by ! x true.
The inverse of the exponential function y ! bx is the logarithmic function x ! by.This function is usually written as y ! logb x.
1. The function is continuous and one-to-one.
Properties of2. The domain is the set of all positive real numbers.
Logarithmic Functions3. The y-axis is an asymptote of the graph.4. The range is the set of all real numbers.5. The graph contains the point (0, 1).
Write an exponential equation equivalent to log3 243 ! 5.35 ! 243
Write a logarithmic equation equivalent to 6"3 ! .
log6 ! %3
Evaluate log8 16.
8&43
&
! 16, so log8 16 ! .
Write each equation in logarithmic form.
1. 27 ! 128 2. 3%4 ! 3. ! "3!
log2 128 ! 7 log3 ! "4 log%17
% ! 3
Write each equation in exponential form.
4. log15 225 ! 2 5. log3 ! %3 6. log4 32 !
152 ! 225 3"3 ! 4%52
%! 32
Evaluate each expression.
7. log4 64 3 8. log2 64 6 9. log100 100,000 2.5
10. log5 625 4 11. log27 81 12. log25 5
13. log2 "7 14. log10 0.00001 "5 15. log4 "2.51&32
1&128
1%2
4%3
1%27
5&2
1&27
1%343
1%81
1&343
1&7
1&81
4&3
1&216
1%216
Example 1Example 1
Example 2Example 2
Example 3Example 3
ExercisesExercises
© Glencoe/McGraw-Hill 580 Glencoe Algebra 2
Solve Logarithmic Equations and Inequalities
Logarithmic to If b # 1, x # 0, and logb x # y, then x # by.Exponential Inequality If b # 1, x # 0, and logb x $ y, then 0 $ x $ by.
Property of Equality for If b is a positive number other than 1, Logarithmic Functions then logb x ! logb y if and only if x ! y.
Property of Inequality for If b # 1, then logb x # logb y if and only if x # y, Logarithmic Functions and logb x $ logb y if and only if x $ y.
Study Guide and Intervention (continued)
Logarithms and Logarithmic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
Solve log2 2x ! 3.log2 2x ! 3 Original equation
2x ! 23 Definition of logarithm
2x ! 8 Simplify.
x ! 4 Simplify.
The solution is x ! 4.
Solve log5 (4x " 3) ' 3.log5 (4x % 3) $ 3 Original equation
0 $ 4x % 3 $ 53 Logarithmic to exponential inequality
3 $ 4x $ 125 ( 3 Addition Property of Inequalities
$ x $ 32 Simplify.
The solution set is 'x | $ x $ 32(.3&4
3&4
Example 1Example 1 Example 2Example 2
ExercisesExercises
Solve each equation or inequality.
1. log2 32 ! 3x 2. log3 2c ! %2
3. log2x 16 ! %2 4. log25 ! " ! 10
5. log4 (5x ( 1) ! 2 3 6. log8 (x % 5) ! 9
7. log4 (3x % 1) ! log4 (2x ( 3) 4 8. log2 (x2 % 6) ! log2 (2x ( 2) 4
9. logx ( 4 27 ! 3 "1 10. log2 (x (3) ! 4 13
11. logx 1000 ! 3 10 12. log8 (4x ( 4) ! 2 15
13. log2 2x # 2 x $ 2 14. log5 x # 2 x $ 25
15. log2 (3x ( 1) $ 4 " ' x ' 5 16. log4 (2x) # % x $
17. log3 (x ( 3) $ 3 "3 ' x ' 24 18. log27 6x # x $ 3%2
2&3
1%4
1&2
1%3
2&3
1&2
x&2
1%8
1%18
5%3
Skills PracticeLogarithms and Logarithmic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
© Glencoe/McGraw-Hill 581 Glencoe Algebra 2
Less
on
10-
2
Write each equation in logarithmic form.
1. 23 ! 8 log2 8 ! 3 2. 32 ! 9 log3 9 ! 2
3. 8%2 ! log8 ! "2 4. ! "2! log%
13
% ! 2
Write each equation in exponential form.
5. log3 243 ! 5 35 ! 243 6. log4 64 ! 3 43 ! 64
7. log9 3 ! 9%12
%! 3 8. log5 ! %2 5"2 !
Evaluate each expression.
9. log5 25 2 10. log9 3
11. log10 1000 3 12. log125 5
13. log4 "3 14. log5 "4
15. log8 83 3 16. log27 "
Solve each equation or inequality. Check your solutions.
17. log3 x ! 5 243 18. log2 x ! 3 8
19. log4 y $ 0 0 ' y ' 1 20. log&14
& x ! 3
21. log2 n # %2 n $ 22. logb 3 ! 9
23. log6 (4x ( 12) ! 2 6 24. log2 (4x % 4) # 5 x $ 9
25. log3 (x ( 2) ! log3 (3x) 1 26. log6 (3y % 5) , log6 (2y ( 3) y ) 8
1&2
1%4
1%64
1%3
1&3
1&625
1&64
1%3
1%2
1%25
1&25
1&2
1%9
1&9
1&3
1%64
1&64
© Glencoe/McGraw-Hill 582 Glencoe Algebra 2
Write each equation in logarithmic form.
1. 53 ! 125 log5 125 ! 3 2. 70 ! 1 log7 1 ! 0 3. 34 ! 81 log3 81 ! 4
4. 3%4 ! 5. ! "3! 6. 7776
&15
&
! 6
log3 ! "4 log%14
% ! 3 log7776 6 !
Write each equation in exponential form.
7. log6 216 ! 3 63 ! 216 8. log2 64 ! 6 26 ! 64 9. log3 ! %4 3"4 !
10. log10 0.00001 ! %5 11. log25 5 ! 12. log32 8 !
10"5 ! 0.00001 25%12
%! 5 32
%35
%! 8
Evaluate each expression.
13. log3 81 4 14. log10 0.0001 "4 15. log2 "4 16. log&13
& 27 "3
17. log9 1 0 18. log8 4 19. log7 "2 20. log6 64 4
21. log3 "1 22. log4 "4 23. log9 9(n ( 1) n # 1 24. 2log2 32 32
Solve each equation or inequality. Check your solutions.
25. log10 n ! %3 26. log4 x # 3 x $ 64 27. log4 x ! 8
28. log&15
& x ! %3 125 29. log7 q $ 0 0 ' q ' 1 30. log6 (2y ( 8) , 2 y ) 14
31. logy 16 ! %4 32. logn ! %3 2 33. logb 1024 ! 5 4
34. log8 (3x ( 7) $ log8 (7x ( 4) 35. log7 (8x ( 20) ! log7 (x ( 6) 36. log3 (x2 % 2) ! log3 x
x $ "2 2
37. SOUND Sounds that reach levels of 130 decibels or more are painful to humans. Whatis the relative intensity of 130 decibels? 1013
38. INVESTING Maria invests $1000 in a savings account that pays 8% interestcompounded annually. The value of the account A at the end of five years can bedetermined from the equation log A ! log[1000(1 ( 0.08)5]. Find the value of A to thenearest dollar. $1469
3%4
1&8
1%2
3&2
1%1000
1&256
1&3
1&49
2%3
1&16
3&5
1&2
1%81
1&81
1%5
1%64
1%81
1&64
1&4
1&81
Practice (Average)
Logarithms and Logarithmic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
Reading to Learn MathematicsLogarithms and Logarithmic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
© Glencoe/McGraw-Hill 583 Glencoe Algebra 2
Less
on
10-
2
Pre-Activity Why is a logarithmic scale used to measure sound?
Read the introduction to Lesson 10-2 at the top of page 531 in your textbook.
How many times louder than a whisper is normal conversation?104 or 10,000 times
Reading the Lesson1. a. Write an exponential equation that is equivalent to log3 81 ! 4. 34 ! 81
b. Write a logarithmic equation that is equivalent to 25%&12
&! . log25 ! "
c. Write an exponential equation that is equivalent to log4 1 ! 0. 40 ! 1
d. Write a logarithmic equation that is equivalent to 10%3 ! 0.001. log10 0.001 ! "3
e. What is the inverse of the function y ! 5x? y ! log5 x
f. What is the inverse of the function y ! log10 x? y ! 10x
2. Match each function with its graph.
a. y ! 3x IV b. y ! log3 x I c. y ! ! "xII
I. II. III.
3. Indicate whether each of the following statements about the exponential function y ! log5 x is true or false.
a. The y-axis is an asymptote of the graph. trueb. The domain is the set of all real numbers. falsec. The graph contains the point (5, 0). falsed. The range is the set of all real numbers. truee. The y-intercept is 1. false
Helping You Remember4. An important skill needed for working with logarithms is changing an equation between
logarithmic and exponential forms. Using the words base, exponent, and logarithm, describean easy way to remember and apply the part of the definition of logarithm that says,“logb x ! y if and only if by ! x.” Sample answer: In these equations, b standsfor base. In log form, b is the subscript, and in exponential form, b is thenumber that is raised to a power. A logarithm is an exponent, so y, which isthe log in the first equation, becomes the exponent in the second equation.
x
y
Ox
y
O
x
y
O
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1%2
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© Glencoe/McGraw-Hill 584 Glencoe Algebra 2
Enrichment
NAME ______________________________________________ DATE______________ PERIOD _____
10-210-2
Musical RelationshipsThe frequencies of notes in a musical scale that are one octave apart arerelated by an exponential equation. For the eight C notes on a piano, theequation is Cn ! C12n " 1, where Cn represents the frequency of note Cn.
1. Find the relationship between C1 and C2.
2. Find the relationship between C1 and C4.
The frequencies of consecutive notes are related by a common ratio r. The general equation is fn ! f1rn " 1.
3. If the frequency of middle C is 261.6 cycles per second and the frequency of the next higher C is 523.2 cycles per second, find the common ratio r. (Hint: The two C’s are 12 notes apart.) Write the answer as a radicalexpression.
4. Substitute decimal values for r and f1 to find a specific equation for fn.
5. Find the frequency of F# above middle C.
6. Frets are a series of ridges placed across the fingerboard of a guitar. Theyare spaced so that the sound made by pressing a string against one frethas about 1.0595 times the wavelength of the sound made by using thenext fret. The general equation is wn ! w0(1.0595)n. Describe thearrangement of the frets on a guitar.
Study Guide and InterventionProperties of Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-310-3
© Glencoe/McGraw-Hill 585 Glencoe Algebra 2
Less
on
10-
3
Properties of Logarithms Properties of exponents can be used to develop thefollowing properties of logarithms.
Product Property For all positive numbers m, n, and b, where b # 1, of Logarithms logb mn ! logb m $ logb n.
Quotient Property For all positive numbers m, n, and b, where b # 1, of Logarithms logb %
mn% ! logb m " logb n.
Power Property For any real number p and positive numbers m and b, of Logarithms where b # 1, logb mp ! p logb m.
Use log3 28 ! 3.0331 and log3 4 ! 1.2619 to approximate the value of each expression.ExampleExample
a. log3 36
log3 36 ! log3 (32 & 4)! log3 32 $ log3 4! 2 $ log3 4! 2 $ 1.2619! 3.2619
b. log3 7
log3 7 ! log3 " #! log3 28 " log3 4! 3.0331 " 1.2619! 1.7712
c. log3 256
log3 256 ! log3 (44)! 4 & log3 4! 4(1.2619)! 5.0476
28%4
ExercisesExercises
Use log12 3 ! 0.4421 and log12 7 ! 0.7831 to evaluate each expression.
1. log12 21 1.2252 2. log12 0.3410 3. log12 49 1.5662
4. log12 36 1.4421 5. log12 63 1.6673 6. log12 !0.2399
7. log12 0.2022 8. log12 16,807 3.9155 9. log12 441 2.4504
Use log5 3 ! 0.6826 and log5 4 ! 0.8614 to evaluate each expression.
10. log5 12 1.5440 11. log5 100 2.8614 12. log5 0.75 !0.1788
13. log5 144 3.0880 14. log5 0.3250 15. log5 375 3.6826
16. log5 1.3$ 0.1788 17. log5 !0.3576 18. log5 1.730481%5
9%16
27%16
81%49
27%49
7%3
© Glencoe/McGraw-Hill 586 Glencoe Algebra 2
Solve Logarithmic Equations You can use the properties of logarithms to solveequations involving logarithms.
Solve each equation.
a. 2 log3 x ! log3 4 " log3 25
2 log3 x ! log3 4 " log3 25 Original equation
log3 x2 ! log3 4 " log3 25 Power Property
log3 " log3 25 Quotient Property
" 25 Property of Equality for Logarithmic Functions
x2 " 100 Multiply each side by 4.
x " #10 Take the square root of each side.
Since logarithms are undefined for x $ 0, !10 is an extraneous solution.The only solution is 10.
b. log2 x % log2 (x % 2) " 3
log2 x % log2 (x % 2) " 3 Original equation
log2 x(x % 2) " 3 Product Property
x(x % 2) " 23 Definition of logarithm
x2 % 2x " 8 Distributive Property
x2 % 2x ! 8 " 0 Subtract 8 from each side.
(x % 4)(x ! 2) " 0 Factor.
x " 2 or x " !4 Zero Product Property
Since logarithms are undefined for x $ 0, !4 is an extraneous solution.The only solution is 2.
Solve each equation. Check your solutions.
1. log5 4 % log5 2x " log5 24 3 2. 3 log4 6 ! log4 8 " log4 x 27
3. log6 25 % log6 x " log6 20 4 4. log2 4 ! log2 (x % 3) " log2 8 !
5. log6 2x ! log6 3 " log6 (x ! 1) 3 6. 2 log4 (x % 1) " log4 (11 ! x) 2
7. log2 x ! 3 log2 5 " 2 log2 10 12,500 8. 3 log2 x ! 2 log2 5x " 2 100
9. log3 (c % 3) ! log3 (4c ! 1) " log3 5 10. log5 (x % 3) ! log5 (2x ! 1) " 24"7
8"19
5"2
1&2
x2&4
x2&4
Study Guide and Intervention (continued)
Properties of Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-310-3
ExampleExample
ExercisesExercises
Skills PracticeProperties of Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-310-3
© Glencoe/McGraw-Hill 587 Glencoe Algebra 2
Less
on
10-
3
Use log2 3 ! 1.5850 and log2 5 ! 2.3219 to approximate the value of eachexpression.
1. log2 25 4.6438 2. log2 27 4.755
3. log2 !0.7369 4. log2 0.7369
5. log2 15 3.9069 6. log2 45 5.4919
7. log2 75 6.2288 8. log2 0.6 !0.7369
9. log2 !1.5850 10. log2 0.8481
Solve each equation. Check your solutions.
11. log10 27 " 3 log10 x 3 12. 3 log7 4 " 2 log7 b 8
13. log4 5 % log4 x " log4 60 12 14. log6 2c % log6 8 " log6 80 5
15. log5 y ! log5 8 " log5 1 8 16. log2 q ! log2 3 " log2 7 21
17. log9 4 % 2 log9 5 " log9 w 100 18. 3 log8 2 ! log8 4 " log8 b 2
19. log10 x % log10 (3x ! 5) " log10 2 2 20. log4 x % log4 (2x ! 3) " log4 2 2
21. log3 d % log3 3 " 3 9 22. log10 y ! log10 (2 ! y) " 0 1
23. log2 s % 2 log2 5 " 0 24. log2 (x % 4) ! log2 (x ! 3) " 3 4
25. log4 (n % 1) ! log4 (n ! 2) " 1 3 26. log5 10 % log5 12 " 3 log5 2 % log5 a 15
1"25
9&5
1&3
5&3
3&5
© Glencoe/McGraw-Hill 588 Glencoe Algebra 2
Use log10 5 ! 0.6990 and log10 7 ! 0.8451 to approximate the value of eachexpression.
1. log10 35 1.5441 2. log10 25 1.3980 3. log10 0.1461 4. log10 !0.1461
5. log10 245 2.3892 6. log10 175 2.2431 7. log10 0.2 !0.6990 8. log10 0.5529
Solve each equation. Check your solutions.
9. log7 n ! log7 8 4 10. log10 u ! log10 4 8
11. log6 x " log6 9 ! log6 54 6 12. log8 48 # log8 w ! log8 4 12
13. log9 (3u " 14) # log9 5 ! log9 2u 2 14. 4 log2 x " log2 5 ! log2 405 3
15. log3 y ! #log3 16 " log3 64 16. log2 d ! 5 log2 2 # log2 8 4
17. log10 (3m # 5) " log10 m ! log10 2 2 18. log10 (b " 3) " log10 b ! log10 4 1
19. log8 (t " 10) # log8 (t # 1) ! log8 12 2 20. log3 (a " 3) " log3 (a " 2) ! log3 6 0
21. log10 (r " 4) # log10 r ! log10 (r " 1) 2 22. log4 (x2 # 4) # log4 (x " 2) ! log4 1 3
23. log10 4 " log10 w ! 2 25 24. log8 (n # 3) " log8 (n " 4) ! 1 4
25. 3 log5 (x2 " 9) # 6 ! 0 "4 26. log16 (9x " 5) # log16 (x2 # 1) ! 3
27. log6 (2x # 5) " 1 ! log6 (7x " 10) 8 28. log2 (5y " 2) # 1 ! log2 (1 # 2y) 0
29. log10 (c2 # 1) # 2 ! log10 (c " 1) 101 30. log7 x " 2 log7 x # log7 3 ! log7 72 6
31. SOUND The loudness L of a sound in decibels is given by L ! 10 log10 R, where R is thesound’s relative intensity. If the intensity of a certain sound is tripled, by how manydecibels does the sound increase? about 4.8 db
32. EARTHQUAKES An earthquake rated at 3.5 on the Richter scale is felt by many people,and an earthquake rated at 4.5 may cause local damage. The Richter scale magnitudereading m is given by m ! log10 x, where x represents the amplitude of the seismic wavecausing ground motion. How many times greater is the amplitude of an earthquake thatmeasures 4.5 on the Richter scale than one that measures 3.5? 10 times
1$2
1#4
1$3
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25$7
5$7
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Practice (Average)
Properties of Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-310-3
Reading to Learn MathematicsProperties of Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-310-3
© Glencoe/McGraw-Hill 589 Glencoe Algebra 2
Less
on
10-
3
Pre-Activity How are the properties of exponents and logarithms related?
Read the introduction to Lesson 10-3 at the top of page 541 in your textbook.Find the value of log5 125. 3 Find the value of log5 5. 1Find the value of log5 (125 % 5). 2Which of the following statements is true? BA. log5 (125 % 5) ! (log5 125) % (log5 5)B. log5 (125 % 5) ! log5 125 # log5 5
Reading the Lesson1. Each of the properties of logarithms can be stated in words or in symbols. Complete the
statements of these properties in words.
a. The logarithm of a quotient is the of the logarithms of the
and the .
b. The logarithm of a power is the of the logarithm of the base and
the .
c. The logarithm of a product is the of the logarithms of its
.
2. State whether each of the following equations is true or false. If the statement is true,name the property of logarithms that is illustrated.
a. log3 10 ! log3 30 # log3 3 true; Quotient Propertyb. log4 12 ! log4 4 " log4 8 falsec. log2 81 ! 2 log2 9 true; Power Propertyd. log8 30 ! log8 5 & log8 6 false
3. The algebraic process of solving the equation log2 x " log2 (x " 2) ! 3 leads to “x ! #4or x ! 2.” Does this mean that both #4 and 2 are solutions of the logarithmic equation?Explain your reasoning. Sample answer: No; 2 is a solution because it checks: log2 2 $ log2 (2 $ 2) % log2 2 $ log2 4 % 1 $ 2 % 3. However,because log2 (!4) and log2 (! 2) are undefined, !4 is an extraneoussolution and must be eliminated. The only solution is 2.
Helping You Remember4. A good way to remember something is to relate it something you already know. Use words
to explain how the Product Property for exponents can help you remember the productproperty for logarithms. Sample answer: When you multiply two numbers orexpressions with the same base, you add the exponents and keep thesame base. Logarithms are exponents, so to find the logarithm of aproduct, you add the logarithms of the factors, keeping the same base.
factorssum
exponentproduct
denominatornumeratordifference
© Glencoe/McGraw-Hill 590 Glencoe Algebra 2
SpiralsConsider an angle in standard position with its vertex at a point O called thepole. Its initial side is on a coordinatized axis called the polar axis. A point Pon the terminal side of the angle is named by the polar coordinates (r, !),where r is the directed distance of the point from O and ! is the measure ofthe angle. Graphs in this system may be drawn on polar coordinate papersuch as the kind shown below.
1. Use a calculator to complete the table for log2r " #12!0#.
(Hint: To find ! on a calculator, press 120 r 2 .)
2. Plot the points found in Exercise 1 on the grid above and connect to form a smooth curve.
This type of spiral is called a logarithmic spiral because the angle measures are proportional to the logarithms of the radii.
r 1 2 3 4 5 6 7 8
) LOG!) LOG"
0
10
20
30
40
5060
708090100110
120130
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250260 270 280
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Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
10-310-3
Study Guide and InterventionCommon Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-410-4
© Glencoe/McGraw-Hill 591 Glencoe Algebra 2
Less
on
10-
4
Common Logarithms Base 10 logarithms are called common logarithms. Theexpression log10 x is usually written without the subscript as log x. Use the key onyour calculator to evaluate common logarithms.The relation between exponents and logarithms gives the following identity.
Inverse Property of Logarithms and Exponents 10log x ! x
Evaluate log 50 to four decimal places.Use the LOG key on your calculator. To four decimal places, log 50 ! 1.6990.
Solve 32x ! 1 " 12.32x " 1 ! 12 Original equation
log 32x " 1 ! log 12 Property of Equality for Logarithms
(2x " 1) log 3 ! log 12 Power Property of Logarithms
2x " 1 ! Divide each side by log 3.
2x ! # 1 Subtract 1 from each side.
x ! ! # 1" Multiply each side by .
x # 0.6309
Use a calculator to evaluate each expression to four decimal places.
1. log 18 2. log 39 3. log 1201.2553 1.5911 2.0792
4. log 5.8 5. log 42.3 6. log 0.0030.7634 1.6263 #2.5229
Solve each equation or inequality. Round to four decimal places.
7. 43x ! 12 0.5975 8. 6x " 2 ! 18 #0.3869
9. 54x # 2 ! 120 1.2437 10. 73x # 1 $ 21 {x |x $ 0.8549}
11. 2.4x " 4 ! 30 #0.1150 12. 6.52x $ 200 {x |x $ 1.4153}
13. 3.64x # 1 ! 85.4 1.1180 14. 2x " 5 ! 3x # 2 13.9666
15. 93x ! 45x " 2 #8.1595 16. 6x # 5 ! 27x " 3 #3.6069
1%2
log 12%log 3
1%2
log 12%log 3
log 12%log 3
LOG
ExercisesExercises
Example 1Example 1
Example 2Example 2
© Glencoe/McGraw-Hill 592 Glencoe Algebra 2
Change of Base Formula The following formula is used to change expressions withdifferent logarithmic bases to common logarithm expressions.
Change of Base Formula For all positive numbers a, b, and n, where a & 1 and b & 1, loga n !
Express log8 15 in terms of common logarithms. Then approximateits value to four decimal places.
log8 15 ! Change of Base Formula
# 1.3023 Simplify.
The value of log8 15 is approximately 1.3023.
Express each logarithm in terms of common logarithms. Then approximate itsvalue to four decimal places.
1. log3 16 2. log2 40 3. log5 35
, 2.5237 , 5.3219 , 2.2091
4. log4 22 5. log12 200 6. log2 50
, 2.2297 , 2.1322 , 5.6439
7. log5 0.4 8. log3 2 9. log4 28.5
, #0.5693 , 0.6309 , 2.4164
10. log3 (20)2 11. log6 (5)4 12. log8 (4)5
, 5.4537 , 3.5930 , 3.3333
13. log5 (8)3 14. log2 (3.6)6 15. log12 (10.5)4
, 3.8761 , 11.0880 , 3.7851
16. log3 $150% 17. log43$39% 18. log5
4$1600%
, 2.2804 , 0.8809 , 1.1460log 1600%%4 log 5
log 39%3 log 4
log 150%2 log 3
4 log 10.5%%log 12
6 log 3.6%%log 2
3 log 8%log 5
5 log 4%log 8
4 log 5%log 6
2 log 20%%log 3
log 28.5%%log 4
log 2%log 3
log 0.4%log 5
log 50%log 2
log 200%log 12
log 22%log 4
log 35%log 5
log 40%log 2
log 16%log 3
log10 15%log10 8
logb n%logb a
Study Guide and Intervention (continued)
Common Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-410-4
ExampleExample
ExercisesExercises
Skills PracticeCommon Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-410-4
© Glencoe/McGraw-Hill 593 Glencoe Algebra 2
Less
on
10-
4
Use a calculator to evaluate each expression to four decimal places.
1. log 6 0.7782 2. log 15 1.1761
3. log 1.1 0.0414 4. log 0.3 #0.5229
Use the formula pH " #log[H!] to find the pH of each substance given itsconcentration of hydrogen ions.
5. gastric juices: [H"] ! 1.0 ' 10#1 mole per liter 1.0
6. tomato juice: [H"] ! 7.94 ' 10#5 mole per liter 4.1
7. blood: [H"] ! 3.98 ' 10#8 mole per liter 7.4
8. toothpaste: [H"] ! 1.26 ' 10#10 mole per liter 9.9
Solve each equation or inequality. Round to four decimal places.
9. 3x ( 243 {x |x & 5} 10. 16v ) !v "v ' # #11. 8p ! 50 1.8813 12. 7y ! 15 1.3917
13. 53b ! 106 0.9659 14. 45k ! 37 0.5209
15. 127p ! 120 0.2752 16. 92m ! 27 0.75
17. 3r # 5 ! 4.1 6.2843 18. 8y " 4 ( 15 {y |y & #2.6977}
19. 7.6d " 3 ! 57.2 #1.0048 20. 0.5t # 8 ! 16.3 3.9732
21. 42x2! 84 (1.0888 22. 5x2 " 1! 10 (0.6563
Express each logarithm in terms of common logarithms. Then approximate itsvalue to four decimal places.
23. log3 7 ; 1.7712 24. log5 66 ; 2.6032
25. log2 35 ; 5.1293 26. log6 10 ; 1.2851log10 10%%log10 6
log10 35%%log10 2
log10 66%%log10 5
log10 7%log10 3
1%2
1%4
© Glencoe/McGraw-Hill 594 Glencoe Algebra 2
Use a calculator to evaluate each expression to four decimal places.
1. log 101 2.0043 2. log 2.2 0.3424 3. log 0.05 #1.3010
Use the formula pH " #log[H!] to find the pH of each substance given itsconcentration of hydrogen ions.
4. milk: [H"] ! 2.51 ' 10#7 mole per liter 6.6
5. acid rain: [H"] ! 2.51 ' 10#6 mole per liter 5.6
6. black coffee: [H"] ! 1.0 ' 10#5 mole per liter 5.0
7. milk of magnesia: [H"] ! 3.16 ' 10#11 mole per liter 10.5
Solve each equation or inequality. Round to four decimal places.
8. 2x * 25 {x |x ) 4.6439} 9. 5a ! 120 2.9746 10. 6z ! 45.6 2.1319
11. 9m $ 100 {m |m $ 2.0959} 12. 3.5x ! 47.9 3.0885 13. 8.2y ! 64.5 1.9802
14. 2b " 1 ) 7.31 {b |b ' 1.8699} 15. 42x ! 27 1.1887 16. 2a # 4 ! 82.1 10.3593
17. 9z # 2 ( 38 {z |z & 3.6555} 18. 5w " 3 ! 17 #1.2396 19. 30x2! 50 (1.0725
20. 5x2 # 3 ! 72 (2.3785 21. 42x ! 9x " 1 3.8188 22. 2n " 1 ! 52n # 1 0.9117
Express each logarithm in terms of common logarithms. Then approximate itsvalue to four decimal places.
23. log5 12 ; 1.5440 24. log8 32 ; 1.6667 25. log11 9 ; 0.9163
26. log2 18 ; 4.1699 27. log9 6 ; 0.8155 28. log7 $8% ;
29. HORTICULTURE Siberian irises flourish when the concentration of hydrogen ions [H"]in the soil is not less than 1.58 ' 10#8 mole per liter. What is the pH of the soil in whichthese irises will flourish? 7.8 or less
30. ACIDITY The pH of vinegar is 2.9 and the pH of milk is 6.6. How many times greater isthe hydrogen ion concentration of vinegar than of milk? about 5000
31. BIOLOGY There are initially 1000 bacteria in a culture. The number of bacteria doubleseach hour. The number of bacteria N present after t hours is N ! 1000(2) t. How long willit take the culture to increase to 50,000 bacteria? about 5.6 h
32. SOUND An equation for loudness L in decibels is given by L ! 10 log R, where R is thesound’s relative intensity. An air-raid siren can reach 150 decibels and jet engine noisecan reach 120 decibels. How many times greater is the relative intensity of the air-raidsiren than that of the jet engine noise? 1000
log10 8%2 log10 7
log10 6%%log10 9
log10 18%%log10 2
log10 9%%log10 11
log10 32%%log10 8
log10 12%%log10 5
Practice (Average)
Common Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-410-4
0.5343
Reading to Learn MathematicsCommon Logarithms
NAME ______________________________________________ DATE ____________ PERIOD _____
10-410-4
© Glencoe/McGraw-Hill 595 Glencoe Algebra 2
Less
on
10-
4
Pre-Activity Why is a logarithmic scale used to measure acidity?
Read the introduction to Lesson 10-4 at the top of page 547 in your textbook.
Which substance is more acidic, milk or tomatoes? tomatoes
Reading the Lesson
1. Rhonda used the following keystrokes to enter an expression on her graphing calculator:
17
The calculator returned the result 1.230448921.Which of the following conclusions are correct? a, c, and d
a. The base 10 logarithm of 17 is about 1.2304.
b. The base 17 logarithm of 10 is about 1.2304.
c. The common logarithm of 17 is about 1.230449.
d. 101.230448921 is very close to 17.
e. The common logarithm of 17 is exactly 1.230448921.
2. Match each expression from the first column with an expression from the second columnthat has the same value.
a. log2 2 iv i. log4 1
b. log 12 iii ii. log2 8
c. log3 1 i iii. log10 12
d. log5 v iv. log5 5
e. log 1000 ii v. log 0.1
3. Calculators do not have keys for finding base 8 logarithms directly. However, you can use
a calculator to find log8 20 if you apply the formula.
Which of the following expressions are equal to log8 20? B and C
A. log20 8 B. C. D.
Helping You Remember
4. Sometimes it is easier to remember a formula if you can state it in words. State thechange of base formula in words. Sample answer: To change the logarithm of anumber from one base to another, divide the log of the original numberin the old base by the log of the new base in the old base.
log 8!log 20
log 20!log 8
log10 20!log10 8
change of base
1!5
ENTER) LOG
© Glencoe/McGraw-Hill 596 Glencoe Algebra 2
The Slide RuleBefore the invention of electronic calculators, computations were oftenperformed on a slide rule. A slide rule is based on the idea of logarithms. It hastwo movable rods labeled with C and D scales. Each of the scales is logarithmic.
To multiply 2 ! 3 on a slide rule, move the C rod to the right as shownbelow. You can find 2 ! 3 by adding log 2 to log 3, and the slide rule adds thelengths for you. The distance you get is 0.778, or the logarithm of 6.
Follow the steps to make a slide rule.
1. Use graph paper that has small squares, such as 10 squares to the inch. Using the scales shown at the right, plot the curve y " log x for x " 1, 1.5,and the whole numbers from 2 through 10. Make an obvious heavy dot for each point plotted.
2. You will need two strips of cardboard. A 5-by-7 index card, cut in half the long way,will work fine. Turn the graph you made in Exercise 1 sideways and use it to marka logarithmic scale on each of the twostrips. The figure shows the mark for 2 being drawn.
3. Explain how to use a slide rule to divide 8 by 2.
0
0.1
0.2
0.3 y
12
1 1.5 2
y = log x
0.1
0.2
1 2
1
21
CD
2
4
3
6
4 5 6 7 8 9
83 5 7 9
log 6
log 3log 2
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
C
D
Enrichment
NAME ______________________________________________ DATE______________ PERIOD _____
10-410-4
Study Guide and InterventionBase e and Natural Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-510-5
© Glencoe/McGraw-Hill 597 Glencoe Algebra 2
Less
on
10-
5
Base e and Natural Logarithms The irrational number e ! 2.71828… often occursas the base for exponential and logarithmic functions that describe real-world phenomena.
Natural Base e As n increases, "1 # #napproaches e ! 2.71828….
ln x " loge x
The functions y " ex and y " ln x are inverse functions.
Inverse Property of Base e and Natural Logarithms eln x " x ln ex " x
Natural base expressions can be evaluated using the ex and ln keys on your calculator.
Evaluate ln 1685.Use a calculator.ln 1685 ! 7.4295
Write a logarithmic equation equivalent to e2x ! 7.e2x " 7 → loge 7 " 2x or 2x " ln 7
Evaluate ln e18.Use the Inverse Property of Base e and Natural Logarithms.ln e18 " 18
Use a calculator to evaluate each expression to four decimal places.
1. ln 732 2. ln 84,350 3. ln 0.735 4. ln 1006.5958 11.3427 "0.3079 4.6052
5. ln 0.0824 6. ln 2.388 7. ln 128,245 8. ln 0.00614"2.4962 0.8705 11.7617 "5.0929
Write an equivalent exponential or logarithmic equation.
9. e15 " x 10. e3x " 45 11. ln 20 " x 12. ln x " 8ln x ! 15 3x ! ln 45 ex ! 20 x ! e8
13. e$5x " 0.2 14. ln (4x) " 9.6 15. e8.2 " 10x 16. ln 0.0002 " x"5x ! ln 0.2 4x ! e9.6 ln 10x ! 8.2 ex ! 0.0002
Evaluate each expression.
17. ln e3 18. eln 42 19. eln 0.5 20. ln e16.2
3 42 0.5 16.2
1%n
Example 1Example 1
Example 2Example 2
Example 3Example 3
ExercisesExercises
© Glencoe/McGraw-Hill 598 Glencoe Algebra 2
Equations and Inequalities with e and ln All properties of logarithms fromearlier lessons can be used to solve equations and inequalities with natural logarithms.
Solve each equation or inequality.
a. 3e2x # 2 " 103e2x # 2 " 10 Original equation
3e2x " 8 Subtract 2 from each side.
e2x " Divide each side by 3.
ln e2x " ln Property of Equality for Logarithms
2x " ln Inverse Property of Exponents and Logarithms
x " ln Multiply each side by %12%.
x ! 0.4904 Use a calculator.
b. ln (4x $ 1) & 2
ln (4x $ 1) & 2 Original inequality
eln (4x $ 1) & e2 Write each side using exponents and base e.
0 & 4x $ 1 & e2 Inverse Property of Exponents and Logarithms
1 & 4x & e2 # 1 Addition Property of Inequalities
& x & (e2 # 1) Multiplication Property of Inequalities
0.25 & x & 2.0973 Use a calculator.
Solve each equation or inequality.
1. e4x " 120 2. ex ' 25 3. ex $ 2 # 4 " 211.1969 {x|x # 3.2189} 4.8332
4. ln 6x ( 4 5. ln (x # 3) $ 5 " $2 6. e$8x ' 50x $ 9.0997 17.0855 {x |x $ "0.4890}
7. e4x $ 1 $ 3 " 12 8. ln (5x # 3) " 3.6 9. 2e3x # 5 " 20.9270 6.7196 no solution
10. 6 # 3ex # 1 " 21 11. ln (2x $ 5) " 8 12. ln 5x # ln 3x ) 90.6094 1492.9790 {x |x % 23.2423}
1%4
1%4
8%3
1%2
8%3
8%3
8%3
Study Guide and Intervention (continued)
Base e and Natural Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-510-5
ExampleExample
ExercisesExercises
NAME ______________________________________________ DATE______________ PERIOD _____
10-510-5
© Glencoe/McGraw-Hill 599 Glencoe Algebra 2
Less
on
10-
5
Use a calculator to evaluate each expression to four decimal places.
1. e3 20.0855 2. e$2 0.1353
3. ln 2 0.6931 4. ln 0.09 "2.4079
Write an equivalent exponential or logarithmic equation.
5. ex " 3 x ! ln 3 6. e4 " 8x 4 ! ln 8x
7. ln 15 " x ex ! 15 8. ln x ! 0.6931 x ! e0.6931
Evaluate each expression.
9. eln 3 3 10. eln 2x 2x
11. ln e$2.5 "2.5 12. ln ey y
Solve each equation or inequality.
13. ex ( 5 {x |x $ 1.6094} 14. ex & 3.2 {x |x & 1.1632}
15. 2ex $ 1 " 11 1.7918 16. 5ex # 3 " 18 1.0986
17. e3x " 30 1.1337 18. e$4x ) 10 {x |x & "0.5756}
19. e5x # 4 ) 34 {x |x % 0.6802} 20. 1 $ 2e2x " $19 1.1513
21. ln 3x " 2 2.4630 22. ln 8x " 3 2.5107
23. ln (x $ 2) " 2 9.3891 24. ln (x # 3) " 1 "0.2817
25. ln (x # 3) " 4 51.5982 26. ln x # ln 2x " 2 1.9221
Skills PracticeBase e and Natural Logarithms
© Glencoe/McGraw-Hill 600 Glencoe Algebra 2
Use a calculator to evaluate each expression to four decimal places.
1. e1.5 4.4817 2. ln 8 2.0794 3. ln 3.2 1.1632 4. e$0.6 0.5488
5. e4.2 66.6863 6. ln 1 0 7. e$2.5 0.0821 8. ln 0.037 "3.2968
Write an equivalent exponential or logarithmic equation.
9. ln 50 " x 10. ln 36 " 2x 11. ln 6 ! 1.7918 12. ln 9.3 ! 2.2300
ex ! 50 e2x ! 36 e1.7918 ! 6 e2.2300 ! 9.3
13. ex " 8 14. e5 " 10x 15. e$x " 4 16. e2 " x # 1
x ! ln 8 5 ! ln 10x x ! "ln 4 2 ! ln (x ' 1)
Evaluate each expression.
17. eln 12 12 18. eln 3x 3x 19. ln e$1 "1 20. ln e$2y "2y
Solve each equation or inequality.
21. ex & 9 22. e$x " 31 23. ex " 1.1 24. ex " 5.8
{x |x & 2.1972} "3.4340 0.0953 1.7579
25. 2ex $ 3 " 1 26. 5ex # 1 ( 7 27. 4 # ex " 19 28. $3ex # 10 & 8
0.6931 {x |x $ 0.1823} 2.7081 {x |x % "0.4055}
29. e3x " 8 30. e$4x " 5 31. e0.5x " 6 32. 2e5x " 24
0.6931 "0.4024 3.5835 0.4970
33. e2x # 1 " 55 34. e3x $ 5 " 32 35. 9 # e2x " 10 36. e$3x # 7 ( 15
1.9945 1.2036 0 {x |x # "0.6931}
37. ln 4x " 3 38. ln ($2x) " 7 39. ln 2.5x " 10 40. ln (x $ 6) " 1
5.0214 "548.3166 8810.5863 8.7183
41. ln (x # 2) " 3 42. ln (x # 3) " 5 43. ln 3x # ln 2x " 9 44. ln 5x # ln x " 7
18.0855 145.4132 36.7493 14.8097
INVESTING For Exercises 45 and 46, use the formula for continuouslycompounded interest, A ! Pert, where P is the principal, r is the annual interestrate, and t is the time in years.
45. If Sarita deposits $1000 in an account paying 3.4% annual interest compoundedcontinuously, what is the balance in the account after 5 years? $1185.30
46. How long will it take the balance in Sarita’s account to reach $2000? about 20.4 yr
47. RADIOACTIVE DECAY The amount of a radioactive substance y that remains after t years is given by the equation y " aekt, where a is the initial amount present and k isthe decay constant for the radioactive substance. If a " 100, y " 50, and k " $0.035,find t. about 19.8 yr
Practice (Average)
Base e and Natural Logarithms
NAME ______________________________________________ DATE______________ PERIOD _____
10-510-5
Reading to Learn MathematicsBase e and Natural Logarithms
NAME ______________________________________________ DATE ____________ PERIOD _____
10-510-5
© Glencoe/McGraw-Hill 601 Glencoe Algebra 2
Less
on
10-
5
Pre-Activity How is the natural base e used in banking?
Read the introduction to Lesson 10-5 at the top of page 554 in your textbook.
Suppose that you deposit $675 in a savings account that pays an annualinterest rate of 5%. In each case listed below, indicate which method ofcompounding would result in more money in your account at the end of oneyear.a. annual compounding or monthly compounding monthlyb. quarterly compounding or daily compounding dailyc. daily compounding or continuous compounding continuous
Reading the Lesson1. Jagdish entered the following keystrokes in his calculator:
5
The calculator returned the result 1.609437912. Which of the following conclusions arecorrect? d and fa. The common logarithm of 5 is about 1.6094.
b. The natural logarithm of 5 is exactly 1.609437912.
c. The base 5 logarithm of e is about 1.6094.
d. The natural logarithm of 5 is about 1.609438.
e. 101.609437912 is very close to 5.
f. e1.609437912 is very close to 5.
2. Match each expression from the first column with its value in the second column. Somechoices may be used more than once or not at all.
a. eln 5 IV I. 1
b. ln 1 V II. 10
c. eln e VI III. !1
d. ln e5 IV IV. 5
e. ln e I V. 0
f. ln ! " III VI. e
Helping You Remember
3. A good way to remember something is to explain it to someone else. Suppose that you arestudying with a classmate who is puzzled when asked to evaluate ln e3. How would youexplain to him an easy way to figure this out? Sample answer: ln means naturallog. The natural log of e3 is the power to which you raise e to get e3. Thisis obviously 3.
1"e
ENTER) LN
© Glencoe/McGraw-Hill 602 Glencoe Algebra 2
Approximations for ! and eThe following expression can be used to approximate e. If greater and greatervalues of n are used, the value of the expression approximates e more andmore closely.
!1 ! "n1
""n
Another way to approximate e is to use this infinite sum. The greater thevalue of n, the closer the approximation.
e # 1 ! 1 ! "12" ! "2
1$ 3" ! "2 $
13 $ 4" ! … ! "2 $ 3 $ 4
1$ … $ n" ! …
In a similar manner, % can be approximated using an infinite productdiscovered by the English mathematician John Wallis (1616–1703).
"%2" # "
21" $ "
23" $ "
43" $ "
45" $ "
65" $ "
67" $ … $ "2n
2&n
1" $ "2n2!n
1" …
Solve each problem.
1. Use a calculator with an ex key to find e to 7 decimal places.
2. Use the expression !1 ! "n1
""nto approximate e to 3 decimal places. Use
5, 100, 500, and 7000 as values of n.
3. Use the infinite sum to approximate e to 3 decimal places. Use the whole numbers from 3 through 6 as values of n.
4. Which approximation method approaches the value of e more quickly?
5. Use a calculator with a % key to find % to 7 decimal places.
6. Use the infinite product to approximate % to 3 decimal places. Use the whole numbers from 3 through 6 as values of n.
7. Does the infinite product give good approximations for % quickly?
8. Show that % 4 ! % 5 is equal to e6 to 4 decimal places.
9. Which is larger, e% or % e?
10. The expression x reaches a maximum value at x # e. Use this fact to prove the inequality you found in Exercise 9.
Enrichment
NAME ______________________________________________ DATE______________ PERIOD _____
10-510-5
Study Guide and InterventionExponential Growth and Decay
NAME ______________________________________________ DATE______________ PERIOD _____
10-610-6
© Glencoe/McGraw-Hill 603 Glencoe Algebra 2
Less
on
10-
6Exponential Decay Depreciation of value and radioactive decay are examples ofexponential decay. When a quantity decreases by a fixed percent each time period, theamount of the quantity after t time periods is given by y # a(1 & r)t, where a is the initialamount and r is the percent decrease expressed as a decimal.Another exponential decay model often used by scientists is y # ae&kt, where k is a constant.
CONSUMER PRICES As technology advances, the price of manytechnological devices such as scientific calculators and camcorders goes down.One brand of hand-held organizer sells for $89.
a. If its price decreases by 6% per year, how much will it cost after 5 years?Use the exponential decay model with initial amount $89, percent decrease 0.06, andtime 5 years.y # a(1 & r)t Exponential decay formula
y # 89(1 & 0.06)5 a # 89, r # 0.06, t # 5
y # $65.32After 5 years the price will be $65.32.
b. After how many years will its price be $50?To find when the price will be $50, again use the exponential decay formula and solve for t.
y # a(1 & r)t Exponential decay formula
50 # 89(1 & 0.06)t y # 50, a # 89, r # 0.06
# (0.94)t Divide each side by 89.
log ! " # log (0.94)t Property of Equality for Logarithms
log ! " # t log 0.94 Power Property
t # Divide each side by log 0.94.
t # 9.3The price will be $50 after about 9.3 years.
1. BUSINESS A furniture store is closing out its business. Each week the owner lowersprices by 25%. After how many weeks will the sale price of a $500 item drop below $100?6 weeks
CARBON DATING Use the formula y ! ae"0.00012t, where a is the initial amount ofCarbon-14, t is the number of years ago the animal lived, and y is the remainingamount after t years.
2. How old is a fossil remain that has lost 95% of its Carbon-14? about 25,000 years old
3. How old is a skeleton that has 95% of its Carbon-14 remaining? about 427.5 years old
log !"5809""
""log 0.94
50"89
50"89
50"89
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 604 Glencoe Algebra 2
Exponential Growth Population increase and growth of bacteria colonies are examplesof exponential growth. When a quantity increases by a fixed percent each time period, theamount of that quantity after t time periods is given by y # a(1 ! r)t, where a is the initialamount and r is the percent increase (or rate of growth) expressed as a decimal.Another exponential growth model often used by scientists is y # aekt, where k is a constant.
A computer engineer is hired for a salary of $28,000. If she gets a5% raise each year, after how many years will she be making $50,000 or more?Use the exponential growth model with a # 28,000, y # 50,000, and r # 0.05 and solve for t.
y # a(1 ! r)t Exponential growth formula
50,000 # 28,000(1 ! 0.05)t y # 50,000, a # 28,000, r # 0.05
# (1.05)t Divide each side by 28,000.
log ! " # log (1.05)t Property of Equality of Logarithms
log ! " # t log 1.05 Power Property
t # Divide each side by log 1.05.
t # 11.9 years Use a calculator.
If raises are given annually, she will be making over $50,000 in 12 years.
1. BACTERIA GROWTH A certain strain of bacteria grows from 40 to 326 in 120 minutes.Find k for the growth formula y # aekt, where t is in minutes. about 0.0175
2. INVESTMENT Carl plans to invest $500 at 8.25% interest, compounded continuously.How long will it take for his money to triple? about 14 years
3. SCHOOL POPULATION There are currently 850 students at the high school, whichrepresents full capacity. The town plans an addition to house 400 more students. If the school population grows at 7.8% per year, in how many years will the new additionbe full? about 5 years
4. EXERCISE Hugo begins a walking program by walking mile per day for one week.
Each week thereafter he increases his mileage by 10%. After how many weeks is hewalking more than 5 miles per day? 24 weeks
5. VOCABULARY GROWTH When Emily was 18 months old, she had a 10-wordvocabulary. By the time she was 5 years old (60 months), her vocabulary was 2500 words.If her vocabulary increased at a constant percent per month, what was that increase?about 14%
1"2
log !"5208""
"log 1.05
50"28
50"28
50"28
Study Guide and Intervention (continued)
Exponential Growth and Decay
NAME ______________________________________________ DATE______________ PERIOD _____
10-610-6
ExampleExample
ExercisesExercises
Skills PracticeExponential Growth and Decay
NAME ______________________________________________ DATE______________ PERIOD _____
10-610-6
© Glencoe/McGraw-Hill 605 Glencoe Algebra 2
Less
on
10-
6Solve each problem.
1. FISHING In an over-fished area, the catch of a certain fish is decreasing at an averagerate of 8% per year. If this decline persists, how long will it take for the catch to reachhalf of the amount before the decline? about 8.3 yr
2. INVESTING Alex invests $2000 in an account that has a 6% annual rate of growth. Tothe nearest year, when will the investment be worth $3600? 10 yr
3. POPULATION A current census shows that the population of a city is 3.5 million. Usingthe formula P # aert, find the expected population of the city in 30 years if the growthrate r of the population is 1.5% per year, a represents the current population in millions,and t represents the time in years. about 5.5 million
4. POPULATION The population P in thousands of a city can be modeled by the equationP # 80e0.015t, where t is the time in years. In how many years will the population of thecity be 120,000? about 27 yr
5. BACTERIA How many days will it take a culture of bacteria to increase from 2000 to50,000 if the growth rate per day is 93.2%? about 4.9 days
6. NUCLEAR POWER The element plutonium-239 is highly radioactive. Nuclear reactorscan produce and also use this element. The heat that plutonium-239 emits has helped topower equipment on the moon. If the half-life of plutonium-239 is 24,360 years, what isthe value of k for this element? about 0.00002845
7. DEPRECIATION A Global Positioning Satellite (GPS) system uses satellite informationto locate ground position. Abu’s surveying firm bought a GPS system for $12,500. TheGPS depreciated by a fixed rate of 6% and is now worth $8600. How long ago did Abubuy the GPS system? about 6.0 yr
8. BIOLOGY In a laboratory, an organism grows from 100 to 250 in 8 hours. What is thehourly growth rate in the growth formula y # a(1 ! r) t? about 12.13%
© Glencoe/McGraw-Hill 606 Glencoe Algebra 2
Solve each problem.
1. INVESTING The formula A # P!1 ! "2tgives the value of an investment after t years in
an account that earns an annual interest rate r compounded twice a year. Suppose $500is invested at 6% annual interest compounded twice a year. In how many years will theinvestment be worth $1000? about 11.7 yr
2. BACTERIA How many hours will it take a culture of bacteria to increase from 20 to2000 if the growth rate per hour is 85%? about 7.5 h
3. RADIOACTIVE DECAY A radioactive substance has a half-life of 32 years. Find theconstant k in the decay formula for the substance. about 0.02166
4. DEPRECIATION A piece of machinery valued at $250,000 depreciates at a fixed rate of12% per year. After how many years will the value have depreciated to $100,000?about 7.2 yr
5. INFLATION For Dave to buy a new car comparably equipped to the one he bought 8 yearsago would cost $12,500. Since Dave bought the car, the inflation rate for cars like his hasbeen at an average annual rate of 5.1%. If Dave originally paid $8400 for the car, howlong ago did he buy it? about 8 yr
6. RADIOACTIVE DECAY Cobalt, an element used to make alloys, has several isotopes.One of these, cobalt-60, is radioactive and has a half-life of 5.7 years. Cobalt-60 is used totrace the path of nonradioactive substances in a system. What is the value of k forCobalt-60? about 0.1216
7. WHALES Modern whales appeared 5&10 million years ago. The vertebrae of a whalediscovered by paleontologists contain roughly 0.25% as much carbon-14 as they wouldhave contained when the whale was alive. How long ago did the whale die? Use k # 0.00012. about 50,000 yr
8. POPULATION The population of rabbits in an area is modeled by the growth equationP(t) # 8e0.26t, where P is in thousands and t is in years. How long will it take for thepopulation to reach 25,000? about 4.4 yr
9. DEPRECIATION A computer system depreciates at an average rate of 4% per month. Ifthe value of the computer system was originally $12,000, in how many months is itworth $7350? about 12 mo
10. BIOLOGY In a laboratory, a culture increases from 30 to 195 organisms in 5 hours.What is the hourly growth rate in the growth formula y # a(1 ! r) t? about 45.4%
r"2
Practice (Average)
Exponential Growth and Decay
NAME ______________________________________________ DATE______________ PERIOD _____
10-610-6
Reading to Learn MathematicsExponential Growth and Decay
NAME ______________________________________________ DATE______________ PERIOD _____
10-610-6
© Glencoe/McGraw-Hill 607 Glencoe Algebra 2
Less
on
10-
6Pre-Activity How can you determine the current value of your car?
Read the introduction to Lesson 10-6 at the top of page 560 in your textbook.
• Between which two years shown in the table did the car depreciate bythe greatest amount?between years 0 and 1
• Describe two ways to calculate the value of the car 6 years after it waspurchased. (Do not actually calculate the value.)Sample answer: 1. Multiply $9200.66 by 0.16 and subtract theresult from $9200.66. 2. Multiply $9200.66 by 0.84.
Reading the Lesson
1. State whether each situation is an example of exponential growth or decay.
a. A city had 42,000 residents in 1980 and 128,000 residents in 2000. growth
b. Raul compared the value of his car when he bought it new to the value when hetraded ‘;lpit in six years later. decay
c. A paleontologist compared the amount of carbon-14 in the skeleton of an animalwhen it died to the amount 300 years later. decay
d. Maria deposited $750 in a savings account paying 4.5% annual interest compoundedquarterly. She did not make any withdrawals or further deposits. She compared thebalance in her passbook immediately after she opened the account to the balance 3 years later. growth
2. State whether each equation represents exponential growth or decay.
a. y # 5e0.15t growth b. y # 1000(1 & 0.05) t decay
c. y # 0.3e&1200t decay d. y # 2(1 ! 0.0001) t growth
Helping You Remember
3. Visualizing their graphs is often a good way to remember the difference betweenmathematical equations. How can your knowledge of the graphs of exponential equationsfrom Lesson 10-1 help you to remember that equations of the form y # a(1 ! r) t
represent exponential growth, while equations of the form y # a(1 & r) t representexponential decay?Sample answer: If a # 0, the graph of y ! abx is always increasing if b # 1 and is always decreasing if 0 $ b $ 1. Since r is always a positivenumber, if b ! 1 % r, the base will be greater than 1 and the function willbe increasing (growth), while if b ! 1 " r, the base will be less than 1and the function will be decreasing (decay).
© Glencoe/McGraw-Hill 608 Glencoe Algebra 2
Effective Annual YieldWhen interest is compounded more than once per year, the effective annualyield is higher than the annual interest rate. The effective annual yield, E, isthe interest rate that would give the same amount of interest if the interestwere compounded once per year. If P dollars are invested for one year, thevalue of the investment at the end of the year is A # P(1 ! E). If P dollarsare invested for one year at a nominal rate r compounded n times per year,
the value of the investment at the end of the year is A # P!1 ! "nr
""n. Setting
the amounts equal and solving for E will produce a formula for the effectiveannual yield.
P(1 ! E) # P!1 ! "nr
""n
1 ! E # !1 ! "nr
""n
E # !1 ! "nr
""n& 1
If compounding is continuous, the value of the investment at the end of oneyear is A # Per. Again set the amounts equal and solve for E. A formula forthe effective annual yield under continuous compounding is obtained.
P(1 ! E) # Per
1 ! E # er
E # er & 1
Enrichment
NAME ______________________________________________ DATE______________ PERIOD _____
10-610-6
Find the effectiveannual yield of an investment made at7.5% compounded monthly.r # 0.075n # 12
E # !1 ! "0.
10275""12
& 1 $ 7.76%
Find the effectiveannual yield of an investment made at6.25% compounded continuously.r # 0.0625
E # e0.0625 & 1 $ 6.45%
Example 1Example 1 Example 2Example 2
Find the effective annual yield for each investment.
1. 10% compounded quarterly 2. 8.5% compounded monthly
3. 9.25% compounded continuously 4. 7.75% compounded continuously
5. 6.5% compounded daily (assume a 365-day year)
6. Which investment yields more interest—9% compounded continuously or 9.2% compounded quarterly?
© Glencoe/McGraw-Hill A2 Glencoe Algebra 2
Answers (Lesson 10-1)
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ain
is th
e se
t of a
ll re
al n
umbe
rs.
Exp
onen
tial F
unct
ion
3.T
he x
-axi
s is
the
asym
ptot
e of
the
grap
h.4.
The
ran
ge is
the
set o
f all
posi
tive
num
bers
if a
#0
and
all n
egat
ive
num
bers
if a
$0.
5.T
he g
raph
con
tain
s th
e po
int (
0, a
).
Exp
onen
tial G
row
thIf
a#
0 an
d b
#1,
the
func
tion
y!
abx
repr
esen
ts e
xpon
entia
l gro
wth
.an
d D
ecay
If a
#0
and
0 $
b$
1, th
e fu
nctio
n y
!ab
xre
pres
ents
exp
onen
tial d
ecay
.
Sk
etch
th
e gr
aph
of
y!
0.1(
4)x .
Th
en s
tate
th
e
fun
ctio
n’s
dom
ain
an
d r
ange
.M
ake
a ta
ble
of v
alue
s.C
onne
ct t
he p
oint
s to
for
m a
sm
ooth
cur
ve.
The
dom
ain
of t
he f
unct
ion
is a
ll re
al n
umbe
rs,w
hile
the
ran
ge is
th
e se
t of
all
posi
tive
rea
l num
bers
.
Det
erm
ine
wh
eth
er e
ach
fu
nct
ion
rep
rese
nts
exp
onen
tial
gr
owth
or d
eca
y.a.
y!
0.5(
2)x
b.y
!%
2.8(
2)x
c.y
!1.
1(0.
5)x
expo
nent
ial g
row
th,
neit
her,
sinc
e %
2.8,
expo
nent
ial d
ecay
,sin
cesi
nce
the
base
,2,i
s th
e va
lue
of a
is le
ss
the
base
,0.5
,is
betw
een
grea
ter
than
1th
an 0
.0
and
1
Sk
etch
th
e gr
aph
of
each
fu
nct
ion
.Th
en s
tate
th
e fu
nct
ion
’s d
omai
n a
nd
ran
ge.
1.y
!3(
2)x
2.y
!%
2 !"x
3.y
!0.
25(5
)x
Dom
ain:
all r
eal
Dom
ain:
all r
eal
Dom
ain:
all r
eal
num
bers
;Ran
ge:a
ll nu
mbe
rs;R
ange
:all
num
bers
;Ran
ge:a
llpo
sitiv
e re
al n
umbe
rsne
gativ
e re
al n
umbe
rspo
sitiv
e re
al n
umbe
rsD
eter
min
e w
het
her
eac
h f
un
ctio
n r
epre
sen
ts e
xpon
enti
al g
row
th o
r d
ecay
.
4.y
!0.
3(1.
2)x
grow
th5.
y!
%5 !
"xne
ither
6.y
!3(
10)%
xde
cay
4 & 5
x
y
O
x
y
O
x
y
O
1 & 4
x%
10
12
3
y0.
025
0.1
0.4
1.6
6.4
x
y
O
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill57
4G
lenc
oe A
lgeb
ra 2
Exp
on
enti
al E
qu
atio
ns
and
Ineq
ual
itie
sA
ll th
e pr
oper
ties
of
rati
onal
exp
onen
tsth
at y
ou k
now
als
o ap
ply
to r
eal e
xpon
ents
.Rem
embe
r th
at a
m'
an!
am(
n ,(a
m)n
!am
n ,an
d am
)an
!am
%n .
Pro
pert
y of
Equ
ality
for
If b
is a
pos
itive
num
ber
othe
r th
an 1
,E
xpon
entia
l Fun
ctio
nsth
en b
x!
by
if an
d on
ly if
x!
y.
Pro
pert
y of
Ineq
ualit
y fo
rIf
b#
1
Exp
onen
tial F
unct
ions
then
bx
#b
yif
and
only
if x
#y
and
bx$
by
if an
d on
ly if
x$
y.
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Exp
onen
tial F
unct
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-1
10-1
Sol
ve 4
x"
1!
2x#
5 .4x
%1
!2x
(5
Orig
inal
equ
atio
n
(22 )
x%
1!
2x(
5R
ewrit
e 4
as 2
2 .
2(x
%1)
!x
(5
Pro
p. o
f Ine
qual
ity fo
r E
xpon
entia
lF
unct
ions
2x%
2 !
x(
5D
istr
ibut
ive
Pro
pert
y
x!
7S
ubtr
act x
and
add
2 to
eac
h si
de.
Sol
ve 5
2x"
1$
.
52x
%1
#O
rigin
al in
equa
lity
52x
%1
#5%
3R
ewrit
e as
5%
3 .
2x%
1 #
%3
Pro
p. o
f Ine
qual
ity fo
r E
xpon
entia
l Fun
ctio
ns
2x#
%2
Add
1 to
eac
h si
de.
x#
%1
Div
ide
each
sid
e by
2.
The
sol
utio
n se
t is
{x|x
#%
1}.
1& 12
5
1& 12
5
1% 12
5Ex
ampl
e1Ex
ampl
e1Ex
ampl
e2Ex
ampl
e2
Exer
cises
Exer
cises
Sim
pli
fy e
ach
exp
ress
ion
.
1.(3#
2$ )#2$
2.25
#2$
'12
5#2$
3.(x#
2$ y3#
2$ )#2$
955
!2"
or 3
125!
2"x2
y6
4.(x#
6$ )(x#
5$ )5.
(x#6$ )#
5$6.
(2x*
)(5x3
*)
x!6"
#!
5"x!
30"10
x4&
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.C
hec
k y
our
solu
tion
.
7.32
x%
1!
3x(
23
8.23
x!
4x(
24
9.32
x%
1!
"
10.4
x(
1!
82x
(3
"11
.8x
%2
!12
.252
x!
125x
(2
6
13.4
#x$
!16
#5$
2014
.x#
3$!
36%&&3 4&
615
.x#
2$!
81& #1 8$&
3
16.3
x%
4$
x'
117
.42x
%2
#2x
(1
x$
18.5
2x$
125x
%5
x$
15
19.1
04x
(1
#10
0x%
220
.73x
$49
x221
.82x
%5
$4x
(8
x$
"x
$or
x'
0x
'%3 41 %
3 % 25 % 2
5 % 31 & 27
2 % 31 & 16
7 % 4
1 % 21 & 9
© Glencoe/McGraw-Hill A3 Glencoe Algebra 2
A
Answers (Lesson 10-1)
Skil
ls P
ract
ice
Exp
onen
tial F
unct
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-1
10-1
©G
lenc
oe/M
cGra
w-H
ill57
5G
lenc
oe A
lgeb
ra 2
Lesson 10-1
Sk
etch
th
e gr
aph
of
each
fu
nct
ion
.Th
en s
tate
th
e fu
nct
ion
’s d
omai
n a
nd
ran
ge.
1.y
!3(
2)x
2.y
!2 !
"x
dom
ain:
all r
eal n
umbe
rs;
dom
ain:
all r
eal n
umbe
rs;
rang
e:al
l pos
itive
num
bers
rang
e:al
l pos
itive
num
bers
Det
erm
ine
wh
eth
er e
ach
fu
nct
ion
rep
rese
nts
exp
onen
tial
gro
wth
or d
eca
y.
3.y
!3(
6)x
grow
th4.
y!
2 !"x
deca
y
5.y
!10
%x
deca
y6.
y!
2(2.
5)x
grow
th
Wri
te a
n e
xpon
enti
al f
un
ctio
n w
hos
e gr
aph
pas
ses
thro
ugh
th
e gi
ven
poi
nts
.
7.(0
,1)
and
(%1,
3)y
!#
$x8.
(0,4
) an
d (1
,12)
y!
4(3)
x
9.(0
,3)
and
(%1,
6)y
!3 #
$x10
.(0,
5) a
nd (
1,15
)y
!5(
3)x
11.(
0,0.
1) a
nd (
1,0.
5)y
!0.
1(5)
x12
.(0,
0.2)
and
(1,
1.6)
y!
0.2(
8)x
Sim
pli
fy e
ach
exp
ress
ion
.
13. (3
#3$ )#
3$27
14.(x
#2$ )#
7$x!
14"
15.5
2#3$
'54
#3$
56!
3"16
.x3*
)x*
x2&
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.C
hec
k y
our
solu
tion
.
17.3
x#
9x
$2
18.2
2x(
3!
321
19.4
9x+
x(
"20
.43x
%2
!16
21.3
2x(
5!
27x
522
.27x
!32
x(
334 % 3
1 % 21 & 7
1 % 21 % 3
9 & 10
x
y
Ox
y
O
1 & 2
©G
lenc
oe/M
cGra
w-H
ill57
6G
lenc
oe A
lgeb
ra 2
Sk
etch
th
e gr
aph
of
each
fu
nct
ion
.Th
en s
tate
th
e fu
nct
ion
’s d
omai
n a
nd
ran
ge.
1.y
!1.
5(2)
x2.
y!
4(3)
x3.
y!
3(0.
5)x
dom
ain:
all r
eal
dom
ain:
all r
eal
dom
ain:
all r
eal
num
bers
;ran
ge:a
ll nu
mbe
rs;r
ange
:all
num
bers
;ran
ge:a
ll po
sitiv
e nu
mbe
rspo
sitiv
e nu
mbe
rspo
sitiv
e nu
mbe
rs
Det
erm
ine
wh
eth
er e
ach
fu
nct
ion
rep
rese
nts
exp
onen
tial
gro
wth
or d
eca
y.
4.y
!5(
0.6)
xde
cay
5.y
!0.
1(2)
xgr
owth
6.y
!5
'4%
xde
cay
Wri
te a
n e
xpon
enti
al f
un
ctio
n w
hos
e gr
aph
pas
ses
thro
ugh
th
e gi
ven
poi
nts
.
7.(0
,1)
and
(%1,
4)8.
(0,2
) an
d (1
,10)
9.(0
,%3)
and
(1,
%1.
5)
y!
#$x
y!
2(5)
xy
!"
3(0.
5)x
10.(
0,0.
8) a
nd (
1,1.
6)11
.(0,
%0.
4) a
nd (
2,%
10)
12.(
0,*
) an
d (3
,8*
)
y!
0.8(
2)x
y!
"0.
4(5)
xy
!&
(2)x
Sim
pli
fy e
ach
exp
ress
ion
.
13.(2
#2$ )#
8$16
14.(n
#3$ )#
75$n1
515
.y#
6$'
y5#
6$y
6!6"
16.1
3#6$
'13
#24$
133!
6"17
.n3
)n*
n3
"&
18.1
25#
11$)
5#11$
52!
11"
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.C
hec
k y
our
solu
tion
.
19.3
3x %
5#
81x
$3
20.7
6x!
72x
%20
"5
21.3
6n%
5$
94n
%3
n$
22.9
2x%
1!
27x
(4
1423
.23n
%1
,!
"nn
)24
.164
n%
1!
1282
n(
1
BIO
LOG
YF
or E
xerc
ises
25
and
26,
use
th
e fo
llow
ing
info
rmat
ion
.T
he in
itia
l num
ber
of b
acte
ria
in a
cul
ture
is 1
2,00
0.T
he n
umbe
r af
ter
3 da
ys is
96,
000.
25.W
rite
an
expo
nent
ial f
unct
ion
to m
odel
the
pop
ulat
ion
yof
bac
teri
a af
ter
xda
ys.
y!
12,0
00(2
)x26
.How
man
y ba
cter
ia a
re t
here
aft
er 6
day
s?76
8,00
027
.ED
UC
ATI
ON
A c
olle
ge w
ith
a gr
adua
ting
cla
ss o
f 40
00 s
tude
nts
in t
he y
ear
2002
pred
icts
tha
t it
will
hav
e a
grad
uati
ng c
lass
of 4
862
in 4
yea
rs.W
rite
an
expo
nent
ial
func
tion
to
mod
el t
he n
umbe
r of
stu
dent
s y
in t
he g
radu
atin
g cl
ass
tye
ars
afte
r 20
02.
y!
4000
(1.0
5)t
11 % 21 % 6
1 & 8
1 % 2
1 % 4
x
y
Ox
y
Ox
y
OPra
ctic
e (A
vera
ge)
Exp
onen
tial F
unct
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-1
10-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 10-1)
Rea
din
g t
o L
earn
Math
emati
csE
xpon
entia
l Fun
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-1
10-1
©G
lenc
oe/M
cGra
w-H
ill57
7G
lenc
oe A
lgeb
ra 2
Lesson 10-1
Pre-
Act
ivit
yH
ow d
oes
an e
xpon
enti
al f
un
ctio
n d
escr
ibe
tou
rnam
ent
pla
y?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-1 a
t th
e to
p of
pag
e 52
3 in
you
r te
xtbo
ok.
How
man
y ro
unds
of
play
wou
ld b
e ne
eded
for
a t
ourn
amen
t w
ith
100
play
ers?
7
Rea
din
g t
he
Less
on
1.In
dica
te w
heth
er e
ach
of t
he f
ollo
win
g st
atem
ents
abo
ut t
he e
xpon
enti
al f
unct
ion
y!
10x
is t
rue
or f
alse
.
a.T
he d
omai
n is
the
set
of
all p
osit
ive
real
num
bers
.fa
lse
b.T
he y
-int
erce
pt is
1.
true
c.T
he f
unct
ion
is o
ne-t
o-on
e.tr
ue
d.T
he y
-axi
s is
an
asym
ptot
e of
the
gra
ph.
fals
e
e.T
he r
ange
is t
he s
et o
f al
l rea
l num
bers
.fa
lse
2.D
eter
min
e w
heth
er e
ach
func
tion
rep
rese
nts
expo
nent
ial g
row
thor
dec
ay.
a.y
!0.
2(3)
x .gr
owth
b.y
!3 !
"x .de
cay
c.y
!0.
4(1.
01)x
.gr
owth
3.Su
pply
the
rea
son
for
each
ste
p in
the
fol
low
ing
solu
tion
of
an e
xpon
enti
al e
quat
ion.
92x
%1
!27
xO
rigi
nal e
quat
ion
(32 )
2x%
1!
(33 )
xR
ewri
te e
ach
side
with
a b
ase
of 3
.32
(2x
%1)
!33
xP
ower
of
a P
ower
2(2x
%1)
!3x
Pro
pert
y of
Equ
ality
for
Exp
onen
tial F
unct
ions
4x%
2 !
3xD
istr
ibut
ive
Pro
pert
yx
%2
!0
Sub
trac
t 3x
from
eac
h si
de.
x!
2A
dd 2
to
each
sid
e.
Hel
pin
g Y
ou
Rem
emb
er
4.O
ne w
ay t
o re
mem
ber
that
pol
ynom
ial f
unct
ions
and
exp
onen
tial
fun
ctio
ns a
re d
iffe
rent
is t
o co
ntra
st t
he p
olyn
omia
l fun
ctio
n y
!x2
and
the
expo
nent
ial f
unct
ion
y!
2x.T
ell a
tle
ast
thre
e w
ays
they
are
dif
fere
nt.
Sam
ple
answ
er:I
n y
!x2
,the
var
iabl
e x
is a
bas
e,bu
t in
y!
2x,t
heva
riab
le x
is a
n ex
pone
nt.T
he g
raph
of
y!
x2is
sym
met
ric
with
res
pect
to t
he y
-axi
s,bu
t th
e gr
aph
of y
!2x
is n
ot.T
he g
raph
of y
!x2
touc
hes
the
x-ax
is a
t (0,
0),b
ut th
e gr
aph
of y
!2x
has
the
x-ax
is a
s an
asy
mpt
ote.
You
can
com
pute
the
val
ue o
f y
!x2
men
tally
for
x!
100,
but
you
cann
otco
mpu
te t
he v
alue
of
y!
2xm
enta
lly fo
r x
!10
0.
2 & 5
©G
lenc
oe/M
cGra
w-H
ill57
8G
lenc
oe A
lgeb
ra 2
Find
ing
Sol
utio
ns o
f xy
!yx
Perh
aps
you
have
not
iced
tha
t if
xan
d y
are
inte
rcha
nged
in e
quat
ions
suc
has
x!
yan
d xy
!1,
the
resu
ltin
g eq
uati
on is
equ
ival
ent
to t
he o
rigi
nal
equa
tion
.The
sam
e is
tru
e of
the
equ
atio
n xy
!yx
.How
ever
,fin
ding
solu
tion
s of
xy
!yx
and
draw
ing
its
grap
h is
not
a s
impl
e pr
oces
s.
Sol
ve e
ach
pro
blem
.Ass
um
e th
at x
and
yar
e p
osit
ive
real
nu
mbe
rs.
1.If
a#
0,w
ill (
a,a)
be
a so
luti
on o
f xy
!yx
? Ju
stif
y yo
ur a
nsw
er.
Yes,
sinc
e aa
!aa
mus
t be
tru
e (R
efle
xive
Pro
p.of
Equ
ality
).
2.If
c#
0,d
#0,
and
(c,d
) is
a s
olut
ion
of x
y!
yx,w
ill (
d,c)
als
o be
a s
olut
ion?
Jus
tify
you
r an
swer
.
Yes;
repl
acin
g x
with
d,y
with
cgi
ves
dc
!cd
;but
if (
c,d)
is a
sol
utio
n,cd
!d
c .S
o,by
the
Sym
met
ric
Pro
pert
y of
Equ
ality
,dc
!cd
is t
rue.
3.U
se 2
as
a va
lue
for
yin
xy
!yx
.The
equ
atio
n be
com
es x
2!
2x.
a.F
ind
equa
tion
s fo
r tw
o fu
ncti
ons,
f(x)
and
g(x
) th
at y
ou c
ould
gra
ph t
ofi
nd t
he s
olut
ions
of x
2!
2x.T
hen
grap
h th
e fu
ncti
ons
on a
sep
arat
esh
eet
of g
raph
pap
er.
f(x)
!x2
,g(x
) !
2xS
ee s
tude
nts’
grap
hs.
b.U
se t
he g
raph
you
dre
w f
or p
art
ato
sta
te t
wo
solu
tion
s fo
r x2
!2x
.T
hen
use
thes
e so
luti
ons
to s
tate
tw
o so
luti
ons
for
xy!
yx.
2,4;
(2,2
),(4
,2)
4.In
thi
s ex
erci
se,a
gra
phin
g ca
lcul
ator
will
be
very
hel
pful
.Use
the
te
chni
que
of E
xerc
ise
3 to
com
plet
e th
e ta
bles
bel
ow.T
hen
grap
h xy
!yx
for
posi
tive
val
ues
of x
and
y.If
the
re a
re a
sym
ptot
es,s
how
the
m in
you
rdi
agra
m u
sing
dot
ted
lines
.Not
e th
at in
the
tab
le,s
ome
valu
es o
f yca
llfo
r on
e va
lue
of x
,oth
ers
call
for
two.
x
y
O
xy
44
24
55
1.8
5
88
1.5
8
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-1
10-1 x
y
%1 2%&1 2&
%3 4%&3 4&
11
22
42
33
2.5
3
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
A
Answers (Lesson 10-2)
Stu
dy
Gu
ide
an
d I
nte
rven
tion
Loga
rith
ms
and
Loga
rith
mic
Fun
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-2
10-2
©G
lenc
oe/M
cGra
w-H
ill57
9G
lenc
oe A
lgeb
ra 2
Lesson 10-2
Log
arit
hm
ic F
un
ctio
ns
and
Exp
ress
ion
s
Def
initi
on o
f Lo
gari
thm
Le
t ban
d x
be p
ositi
ve n
umbe
rs, b
"1.
The
loga
rithm
of x
with
bas
e b
is d
enot
ed
with
Bas
e b
log b
xan
d is
def
ined
as
the
expo
nent
yth
at m
akes
the
equa
tion
by!
xtr
ue.
The
inve
rse
of t
he e
xpon
enti
al f
unct
ion
y!
bxis
the
log
arit
hm
ic f
un
ctio
nx
!by
.T
his
func
tion
is u
sual
ly w
ritt
en a
s y
!lo
g bx.
1.T
he fu
nctio
n is
con
tinuo
us a
nd o
ne-t
o-on
e.
Pro
pert
ies
of2.
The
dom
ain
is th
e se
t of a
ll po
sitiv
e re
al n
umbe
rs.
Loga
rith
mic
Fun
ctio
ns3.
The
y-a
xis
is a
n as
ympt
ote
of th
e gr
aph.
4.T
he r
ange
is th
e se
t of a
ll re
al n
umbe
rs.
5.T
he g
raph
con
tain
s th
e po
int (
0, 1
).
Wri
te a
n e
xpon
enti
al e
quat
ion
equ
ival
ent
to l
og3
243
!5.
35!
243
Wri
te a
log
arit
hm
ic e
quat
ion
equ
ival
ent
to 6
"3
!.
log 6
!%
3
Eva
luat
e lo
g 816
.
8&4 3&
!16
,so
log 8
16 !
.
Wri
te e
ach
equ
atio
n i
n l
ogar
ith
mic
for
m.
1.27
!12
82.
3%4
!3.
!"3
!
log 2
128
!7
log 3
!"
4lo
g%1 7%
!3
Wri
te e
ach
equ
atio
n i
n e
xpon
enti
al f
orm
.
4.lo
g 15
225
!2
5.lo
g 3!
%3
6.lo
g 432
!
152
!22
53"
3!
4%5 2%!
32
Eva
luat
e ea
ch e
xpre
ssio
n.
7.lo
g 464
38.
log 2
646
9.lo
g 100
100,
000
2.5
10.l
og5
625
411
.log
2781
12.l
og25
5
13.l
og2
"7
14.l
og10
0.00
001
"5
15.l
og4
"2.
51 & 32
1& 12
8
1 % 24 % 31 % 27
5 & 21 & 27
1% 34
31 % 81
1& 34
31 & 7
1 & 81
4 & 3
1& 21
6
1% 21
6
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exam
ple3
Exam
ple3
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill58
0G
lenc
oe A
lgeb
ra 2
Solv
e Lo
gar
ith
mic
Eq
uat
ion
s an
d In
equ
alit
ies
Loga
rith
mic
to
If b
#1,
x#
0, a
nd lo
g bx
#y,
then
x#
by.
Exp
onen
tial I
nequ
ality
If b
#1,
x#
0, a
nd lo
g bx
$y,
then
0 $
x$
by.
Pro
pert
y of
Equ
ality
for
If b
is a
pos
itive
num
ber
othe
r th
an 1
, Lo
gari
thm
ic F
unct
ions
then
log b
x!
log b
yif
and
only
if x
!y.
Pro
pert
y of
Ineq
ualit
y fo
r If
b#
1, th
en lo
g bx
#lo
g by
if an
d on
ly if
x#
y,
Loga
rith
mic
Fun
ctio
nsan
d lo
g bx
$lo
g by
if an
d on
ly if
x$
y.
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Loga
rith
ms
and
Loga
rith
mic
Fun
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-2
10-2
Sol
ve l
og2
2x!
3.lo
g 22x
!3
Orig
inal
equ
atio
n
2x!
23D
efin
ition
of l
ogar
ithm
2x!
8S
impl
ify.
x!
4S
impl
ify.
The
sol
utio
n is
x!
4.
Sol
ve l
og5
(4x
"3)
'3.
log 5
(4x
%3)
$3
Orig
inal
equ
atio
n
0 $
4x%
3 $
53Lo
garit
hmic
to e
xpon
entia
l ine
qual
ity
3 $
4x$
125
(3
Add
ition
Pro
pert
y of
Ineq
ualit
ies
$x
$32
Sim
plify
.
The
sol
utio
n se
t is
'x|$
x$
32(.
3 & 4
3 & 4
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.
1.lo
g 232
!3x
2.lo
g 32c
!%
2
3.lo
g 2x
16 !
%2
4.lo
g 25!
"!10
5.lo
g 4(5
x(
1) !
23
6.lo
g 8(x
%5)
!9
7.lo
g 4(3
x%
1) !
log 4
(2x
(3)
48.
log 2
(x2
%6)
!lo
g 2(2
x(
2)4
9.lo
g x(
427
!3
"1
10.l
og2
(x(
3) !
413
11.l
ogx
1000
!3
1012
.log
8(4
x(
4) !
215
13.l
og2
2x#
2x
$2
14.l
og5
x#
2x
$25
15.l
og2
(3x
(1)
$4
"'
x'
516
.log
4(2
x) #
%x
$
17.l
og3
(x(
3) $
3"
3 '
x'
2418
.log
276x
#x
$3 % 2
2 & 3
1 % 41 & 2
1 % 3
2 & 3
1 & 2x & 2
1 % 8
1 % 185 % 3
© Glencoe/McGraw-Hill A6 Glencoe Algebra 2
Answers (Lesson 10-2)
Skil
ls P
ract
ice
Loga
rith
ms
and
Loga
rith
mic
Fun
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-2
10-2
©G
lenc
oe/M
cGra
w-H
ill58
1G
lenc
oe A
lgeb
ra 2
Lesson 10-2
Wri
te e
ach
equ
atio
n i
n l
ogar
ith
mic
for
m.
1.23
!8
log 2
8 !
32.
32!
9lo
g 39
!2
3.8%
2!
log 8
!"
24.
!"2
!lo
g %1 3%
!2
Wri
te e
ach
equ
atio
n i
n e
xpon
enti
al f
orm
.
5.lo
g 324
3 !
535
!24
36.
log 4
64 !
343
!64
7.lo
g 93
!9%1 2%
!3
8.lo
g 5!
%2
5"2
!
Eva
luat
e ea
ch e
xpre
ssio
n.
9.lo
g 525
210
.log
93
11.l
og10
1000
312
.log
125
5
13.l
og4
"3
14.l
og5
"4
15.l
og8
833
16.l
og27
"
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.C
hec
k y
our
solu
tion
s.
17.l
og3
x!
524
318
.log
2x
!3
8
19.l
og4
y$
00
'y
'1
20.l
og&1 4&
x!
3
21.l
og2
n#
%2
n$
22.l
ogb
3 !
9
23.l
og6
(4x
(12
) !
26
24.l
og2
(4x
%4)
#5
x$
9
25.l
og3
(x(
2) !
log 3
(3x)
126
.log
6(3
y%
5) ,
log 6
(2y
(3)
y)
8
1 & 21 % 4
1 % 641 % 31 & 31
& 625
1 & 64
1 % 3
1 % 2
1 % 251 & 25
1 & 2
1 % 91 & 9
1 & 31 % 64
1 & 64
©G
lenc
oe/M
cGra
w-H
ill58
2G
lenc
oe A
lgeb
ra 2
Wri
te e
ach
equ
atio
n i
n l
ogar
ith
mic
for
m.
1.53
!12
5lo
g 512
5 !
32.
70!
1lo
g 71
!0
3.34
!81
log 3
81 !
4
4.3%
4!
5.!
"3!
6.77
76&1 5&
!6
log 3
!"
4lo
g %1 4%
!3
log 7
776
6 !
Wri
te e
ach
equ
atio
n i
n e
xpon
enti
al f
orm
.
7.lo
g 621
6 !
363
!21
68.
log 2
64 !
626
!64
9.lo
g 3!
%4
3"4
!
10.l
og10
0.00
001
!%
511
.log
255
!12
.log
328
!
10"
5!
0.00
001
25%1 2%
!5
32%3 5%
!8
Eva
luat
e ea
ch e
xpre
ssio
n.
13.l
og3
814
14.l
og10
0.00
01"
415
.log
2"
416
.log
&1 3&27
"3
17.l
og9
10
18.l
og8
419
.log
7"
220
.log
664
4
21.l
og3
"1
22.l
og4
"4
23.l
og9
9(n
(1)
n#
124
.2lo
g 232
32
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.C
hec
k y
our
solu
tion
s.
25.l
og10
n!
%3
26.l
og4
x#
3x
$64
27.l
og4
x!
8
28.l
og&1 5&
x!
%3
125
29.l
og7
q$
00
'q
'1
30.l
og6
(2y
(8)
,2
y)
14
31.l
ogy
16 !
%4
32.l
ogn
!%
32
33.l
ogb
1024
!5
4
34.l
og8
(3x
(7)
$lo
g 8(7
x(
4)35
.log
7(8
x(
20) !
log 7
(x(
6)36
.log
3(x
2%
2) !
log 3
x
x$
"2
2
37.S
OU
ND
Soun
ds t
hat
reac
h le
vels
of
130
deci
bels
or
mor
e ar
e pa
infu
l to
hum
ans.
Wha
tis
the
rel
ativ
e in
tens
ity
of 1
30 d
ecib
els?
1013
38.I
NV
ESTI
NG
Mar
ia in
vest
s $1
000
in a
sav
ings
acc
ount
tha
t pa
ys 8
% in
tere
stco
mpo
unde
d an
nual
ly.T
he v
alue
of
the
acco
unt
Aat
the
end
of
five
yea
rs c
an b
ede
term
ined
fro
m t
he e
quat
ion
log
A!
log[
1000
(1 (
0.08
)5].
Fin
d th
e va
lue
of A
to t
hene
ares
t do
llar.
$146
9
3 % 4
1 & 81 % 2
3 & 21
% 1000
1& 25
61 & 3
1 & 492 % 3
1 & 16
3 & 51 & 2
1 % 811 & 81
1 % 51 % 64
1 % 81
1 & 641 & 4
1 & 81Pra
ctic
e (A
vera
ge)
Loga
rith
ms
and
Loga
rith
mic
Fun
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-2
10-2
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
A
Answers (Lesson 10-2)
Rea
din
g t
o L
earn
Math
emati
csLo
gari
thm
s an
d Lo
gari
thm
ic F
unct
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-2
10-2
©G
lenc
oe/M
cGra
w-H
ill58
3G
lenc
oe A
lgeb
ra 2
Lesson 10-2
Pre-
Act
ivit
yW
hy
is a
log
arit
hm
ic s
cale
use
d t
o m
easu
re s
oun
d?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-2 a
t th
e to
p of
pag
e 53
1 in
you
r te
xtbo
ok.
How
man
y ti
mes
loud
er t
han
a w
hisp
er is
nor
mal
con
vers
atio
n?10
4or
10,
000
times
Rea
din
g t
he
Less
on
1.a.
Wri
te a
n ex
pone
ntia
l equ
atio
n th
at is
equ
ival
ent
to lo
g 381
!4.
34!
81
b.W
rite
a lo
gari
thm
ic e
quat
ion
that
is e
quiv
alen
t to
25%
&1 2&!
.lo
g 25
!"
c.W
rite
an
expo
nent
ial e
quat
ion
that
is e
quiv
alen
t to
log 4
1 !
0.40
!1
d.W
rite
a lo
gari
thm
ic e
quat
ion
that
is e
quiv
alen
t to
10%
3!
0.00
1.lo
g 10
0.00
1 !
"3
e.W
hat
is t
he in
vers
e of
the
fun
ctio
n y
!5x
?y
!lo
g 5x
f.W
hat
is t
he in
vers
e of
the
fun
ctio
n y
!lo
g 10
x?y
!10
x
2.M
atch
eac
h fu
ncti
on w
ith
its
grap
h.
a.y
!3x
IVb.
y!
log 3
xI
c.y
!!
"xII
I.II
.II
I.
3.In
dica
te w
heth
er e
ach
of t
he f
ollo
win
g st
atem
ents
abo
ut t
he e
xpon
enti
al f
unct
ion
y!
log 5
xis
tru
eor
fal
se.
a.T
he y
-axi
s is
an
asym
ptot
e of
the
gra
ph.
true
b.T
he d
omai
n is
the
set
of
all r
eal n
umbe
rs.
fals
ec.
The
gra
ph c
onta
ins
the
poin
t (5
,0).
fals
ed.
The
ran
ge is
the
set
of
all r
eal n
umbe
rs.
true
e.T
he y
-int
erce
pt is
1.
fals
e
Hel
pin
g Y
ou
Rem
emb
er4.
An
impo
rtan
t sk
ill n
eede
d fo
r w
orki
ng w
ith
loga
rith
ms
is c
hang
ing
an e
quat
ion
betw
een
loga
rith
mic
and
exp
onen
tial
form
s.U
sing
the
wor
ds b
ase,
expo
nent
,and
loga
rith
m,d
escr
ibe
an e
asy
way
to
rem
embe
r an
d ap
ply
the
part
of
the
defi
niti
on o
f lo
gari
thm
tha
t sa
ys,
“log
bx
!y
if a
nd o
nly
if b
y!
x.”
Sam
ple
answ
er:I
n th
ese
equa
tions
,bst
ands
for
base
.In
log
form
,bis
the
sub
scri
pt,a
nd in
exp
onen
tial f
orm
,bis
the
num
ber
that
is r
aise
d to
a p
ower
.A lo
gari
thm
is a
n ex
pone
nt,s
o y,
whi
ch is
the
log
in th
e fir
st e
quat
ion,
beco
mes
the
expo
nent
in th
e se
cond
equ
atio
n.
x
y
Ox
y
O
x
y
O
1 & 3
1 % 21 % 5
1 & 5
©G
lenc
oe/M
cGra
w-H
ill58
4G
lenc
oe A
lgeb
ra 2
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-2
10-2
Mus
ical
Rel
atio
nshi
psT
he f
requ
enci
es o
f no
tes
in a
mus
ical
sca
le t
hat
are
one
octa
ve a
part
are
rela
ted
by a
n ex
pone
ntia
l equ
atio
n.Fo
r th
e ei
ght
C n
otes
on
a pi
ano,
the
equa
tion
is C
n!
C12
n%
1 ,w
here
Cn
repr
esen
ts t
he f
requ
ency
of
note
Cn.
1.F
ind
the
rela
tion
ship
bet
wee
n C
1an
d C
2.C
2!
2C1
2.F
ind
the
rela
tion
ship
bet
wee
n C
1an
d C
4.C
4!
8C1
The
fre
quen
cies
of
cons
ecut
ive
note
s ar
e re
late
d by
a
com
mon
rat
io r
.The
gen
eral
equ
atio
n is
f n!
f 1rn
%1 .
3.If
the
fre
quen
cy o
f m
iddl
e C
is 2
61.6
cyc
les
per
seco
nd
and
the
freq
uenc
y of
the
nex
t hi
gher
C is
523
.2 c
ycle
s pe
r se
cond
,fin
d th
e co
mm
on r
atio
r.(
Hin
t:T
he t
wo
C’s
ar
e 12
not
es a
part
.) W
rite
the
ans
wer
as
a ra
dica
lex
pres
sion
.
r!
12 !2"
4.Su
bsti
tute
dec
imal
val
ues
for
ran
d f 1
to f
ind
a sp
ecif
ic
equa
tion
for
f n.
f n!
261.
1(1.
0594
6)n
"1
5.F
ind
the
freq
uenc
y of
F#
abov
e m
iddl
e C
.
f 7!
261.
6(1.
0594
6)6
%36
9.95
6.F
rets
are
a s
erie
s of
rid
ges
plac
ed a
cros
s th
e fi
nger
boar
d of
a g
uita
r.T
hey
are
spac
ed s
o th
at t
he s
ound
mad
e by
pre
ssin
g a
stri
ng a
gain
st o
ne f
ret
has
abou
t 1.
0595
tim
es t
he w
avel
engt
h of
the
sou
nd m
ade
by u
sing
the
next
fre
t.T
he g
ener
al e
quat
ion
is w
n!
w0(
1.05
95)n
.Des
crib
e th
ear
rang
emen
t of
the
fre
ts o
n a
guit
ar.
The
fret
s ar
e sp
aced
in a
lo
gari
thm
ic s
cale
.
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 10-3)
Stu
dy
Gu
ide
an
d I
nte
rven
tion
Pro
pert
ies
of L
ogar
ithm
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-3
10-3
©G
lenc
oe/M
cGra
w-H
ill58
5G
lenc
oe A
lgeb
ra 2
Lesson 10-3
Pro
per
ties
of
Log
arit
hm
sP
rope
rtie
s of
exp
onen
ts c
an b
e us
ed t
o de
velo
p th
efo
llow
ing
prop
erti
es o
f lo
gari
thm
s.
Pro
duct
Pro
pert
y F
or a
ll po
sitiv
e nu
mbe
rs m
, n, a
nd b
, whe
re b
"1,
of
Log
arith
ms
log b
mn
!lo
g bm
(lo
g bn.
Quo
tient
Pro
pert
y F
or a
ll po
sitiv
e nu
mbe
rs m
, n, a
nd b
, whe
re b
"1,
of
Log
arith
ms
log b
&m n&!
log b
m%
log b
n.
Pow
er P
rope
rty
For
any
rea
l num
ber
pan
d po
sitiv
e nu
mbe
rs m
and
b,
of L
ogar
ithm
sw
here
b"
1, lo
g bm
p!
plo
g bm
.
Use
log
328
)3.
0331
an
d l
og3
4 )
1.26
19 t
o ap
pro
xim
ate
the
valu
e of
eac
h e
xpre
ssio
n.
Exam
ple
Exam
ple
a.lo
g 336
log 3
36!
log 3
(32
'4)
!lo
g 332
(lo
g 34
!2
(lo
g 34
)2
(1.
2619
)3.
2619
b.lo
g 37
log 3
7!
log 3
!"
!lo
g 328
%lo
g 34
)3.
0331
%1.
2619
)1.
7712
c.lo
g 325
6
log 3
256
!lo
g 3(4
4 )!
4 '
log 3
4)
4(1.
2619
))
5.04
76
28 & 4
Exer
cises
Exer
cises
Use
log
123
)0.
4421
an
d l
og12
7 )
0.78
31 t
o ev
alu
ate
each
exp
ress
ion
.
1.lo
g 12
211.
2252
2.lo
g 12
0.34
103.
log 1
249
1.56
62
4.lo
g 12
361.
4421
5.lo
g 12
631.
6673
6.lo
g 12
"0.
2399
7.lo
g 12
0.20
228.
log 1
216
,807
3.91
559.
log 1
244
12.
4504
Use
log
53
)0.
6826
an
d l
og5
4 )
0.86
14 t
o ev
alu
ate
each
exp
ress
ion
.
10.l
og5
121.
5440
11.l
og5
100
2.86
1412
.log
50.
75"
0.17
88
13.l
og5
144
3.08
8014
.log
50.
3250
15.l
og5
375
3.68
26
16.l
og5
1.3$
0.17
8817
.log
5"
0.35
7618
.log
51.
7304
81 & 59 & 1627 & 16
81 & 49
27 & 49
7 & 3
©G
lenc
oe/M
cGra
w-H
ill58
6G
lenc
oe A
lgeb
ra 2
Solv
e Lo
gar
ith
mic
Eq
uat
ion
sYo
u ca
n us
e th
e pr
oper
ties
of
loga
rith
ms
to s
olve
equa
tion
s in
volv
ing
loga
rith
ms.
Sol
ve e
ach
equ
atio
n.
a.2
log 3
x%
log 3
4 !
log 3
25
2 lo
g 3x
%lo
g 34
!lo
g 325
Orig
inal
equ
atio
n
log 3
x2%
log 3
4 !
log 3
25P
ower
Pro
pert
y
log 3
!lo
g 325
Quo
tient
Pro
pert
y
!25
Pro
pert
y of
Equ
ality
for
Loga
rithm
ic F
unct
ions
x2!
100
Mul
tiply
eac
h si
de b
y 4.
x!
-10
Take
the
squa
re r
oot o
f eac
h si
de.
Sinc
e lo
gari
thm
s ar
e un
defi
ned
for
x$
0,%
10 is
an
extr
aneo
us s
olut
ion.
The
onl
y so
luti
on is
10.
b.lo
g 2x
(lo
g 2(x
(2)
!3
log 2
x(
log 2
(x(
2) !
3O
rigin
al e
quat
ion
log 2
x(x
(2)
!3
Pro
duct
Pro
pert
y
x(x
(2)
!23
Def
initi
on o
f log
arith
m
x2(
2x!
8D
istr
ibut
ive
Pro
pert
y
x2(
2x %
8 !
0S
ubtr
act 8
from
eac
h si
de.
(x(
4)(x
%2)
!0
Fac
tor.
x!
2or
x!
%4
Zer
o P
rodu
ct P
rope
rty
Sinc
e lo
gari
thm
s ar
e un
defi
ned
for
x$
0,%
4 is
an
extr
aneo
us s
olut
ion.
The
onl
y so
luti
on is
2.
Sol
ve e
ach
equ
atio
n.C
hec
k y
our
solu
tion
s.
1.lo
g 54
(lo
g 52x
!lo
g 524
32.
3 lo
g 46
%lo
g 48
!lo
g 4x
27
3.lo
g 625
(lo
g 6x
!lo
g 620
44.
log 2
4 %
log 2
(x(
3) !
log 2
8"
5.lo
g 62x
%lo
g 63
!lo
g 6(x
%1)
36.
2 lo
g 4(x
(1)
!lo
g 4(1
1 %
x)2
7.lo
g 2x
%3
log 2
5 !
2 lo
g 210
12,5
008.
3 lo
g 2x
%2
log 2
5x!
210
0
9.lo
g 3(c
(3)
%lo
g 3(4
c%
1) !
log 3
510
.log
5(x
(3)
%lo
g 5(2
x%
1) !
24 % 7
8 % 19
5 % 21 & 2
x2& 4x2& 4
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Pro
pert
ies
of L
ogar
ithm
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-3
10-3
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A9 Glencoe Algebra 2
A
Answers (Lesson 10-3)
Skil
ls P
ract
ice
Pro
pert
ies
of L
ogar
ithm
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-3
10-3
©G
lenc
oe/M
cGra
w-H
ill58
7G
lenc
oe A
lgeb
ra 2
Lesson 10-3
Use
log
23
)1.
5850
an
d l
og2
5 )
2.32
19 t
o ap
pro
xim
ate
the
valu
e of
eac
hex
pre
ssio
n.
1.lo
g 225
4.64
382.
log 2
274.
755
3.lo
g 2"
0.73
694.
log 2
0.73
69
5.lo
g 215
3.90
696.
log 2
455.
4919
7.lo
g 275
6.22
888.
log 2
0.6
"0.
7369
9.lo
g 2"
1.58
5010
.log
20.
8481
Sol
ve e
ach
equ
atio
n.C
hec
k y
our
solu
tion
s.
11.l
og10
27 !
3 lo
g 10
x3
12.3
log 7
4 !
2 lo
g 7b
8
13.l
og4
5 (
log 4
x!
log 4
6012
14.l
og6
2c(
log 6
8 !
log 6
805
15.l
og5
y%
log 5
8 !
log 5
18
16.l
og2
q%
log 2
3 !
log 2
721
17.l
og9
4 (
2 lo
g 95
!lo
g 9w
100
18.3
log 8
2 %
log 8
4 !
log 8
b2
19.l
og10
x(
log 1
0(3
x%
5) !
log 1
02
220
.log
4x
(lo
g 4(2
x%
3) !
log 4
22
21.l
og3
d(
log 3
3 !
39
22.l
og10
y%
log 1
0(2
%y)
!0
1
23.l
og2
s(
2 lo
g 25
!0
24.l
og2
(x(
4) %
log 2
(x%
3) !
34
25.l
og4
(n(
1) %
log 4
(n%
2) !
13
26.l
og5
10 (
log 5
12 !
3 lo
g 52
(lo
g 5a
15
1 % 25
9 & 51 & 3
5 & 33 & 5
©G
lenc
oe/M
cGra
w-H
ill58
8G
lenc
oe A
lgeb
ra 2
Use
log
105
)0.
6990
an
d l
og10
7 )
0.84
51 t
o ap
pro
xim
ate
the
valu
e of
eac
hex
pre
ssio
n.
1.lo
g 10
351.
5441
2.lo
g 10
251.
3980
3.lo
g 10
0.14
614.
log 1
0"
0.14
61
5.lo
g 10
245
2.38
926.
log 1
017
52.
2431
7.lo
g 10
0.2
"0.
6990
8.lo
g 10
0.55
29
Sol
ve e
ach
equ
atio
n.C
hec
k y
our
solu
tion
s.
9.lo
g 7n
!lo
g 78
410
.log
10u
!lo
g 10
48
11.l
og6
x(
log 6
9 !
log 6
546
12.l
og8
48 %
log 8
w!
log 8
412
13.l
og9
(3u
(14
) %
log 9
5 !
log 9
2u2
14.4
log 2
x(
log 2
5 !
log 2
405
3
15.l
og3
y!
%lo
g 316
(lo
g 364
16.l
og2
d!
5 lo
g 22
%lo
g 28
4
17.l
og10
(3m
%5)
(lo
g 10
m!
log 1
02
218
.log
10(b
(3)
(lo
g 10
b!
log 1
04
1
19.l
og8
(t(
10)
%lo
g 8(t
%1)
!lo
g 812
220
.log
3(a
(3)
(lo
g 3(a
(2)
!lo
g 36
0
21.l
og10
(r(
4) %
log 1
0r
!lo
g 10
(r(
1)2
22.l
og4
(x2
%4)
%lo
g 4(x
(2)
!lo
g 41
3
23.l
og10
4 (
log 1
0w
!2
2524
.log
8(n
%3)
(lo
g 8(n
(4)
!1
4
25.3
log 5
(x2
(9)
%6
!0
*4
26.l
og16
(9x
(5)
%lo
g 16
(x2
%1)
!3
27.l
og6
(2x
%5)
(1
!lo
g 6(7
x(
10)
828
.log
2(5
y(
2) %
1 !
log 2
(1 %
2y)
0
29.l
og10
(c2
%1)
%2
!lo
g 10
(c(
1)10
130
.log
7x
(2
log 7
x%
log 7
3 !
log 7
726
31.S
OU
ND
The
loud
ness
Lof
a s
ound
in d
ecib
els
is g
iven
by
L!
10 lo
g 10
R,w
here
Ris
the
soun
d’s
rela
tive
inte
nsit
y.If
the
inte
nsit
y of
a c
erta
in s
ound
is t
ripl
ed,b
y ho
w m
any
deci
bels
doe
s th
e so
und
incr
ease
?ab
out
4.8
db
32.E
ART
HQ
UA
KES
An
eart
hqua
ke r
ated
at
3.5
on t
he R
icht
er s
cale
is f
elt
by m
any
peop
le,
and
an e
arth
quak
e ra
ted
at 4
.5 m
ay c
ause
loca
l dam
age.
The
Ric
hter
sca
le m
agni
tude
read
ing
mis
giv
en b
y m
!lo
g 10
x,w
here
xre
pres
ents
the
am
plit
ude
of t
he s
eism
ic w
ave
caus
ing
grou
nd m
otio
n.H
ow m
any
tim
es g
reat
er is
the
am
plit
ude
of a
n ea
rthq
uake
tha
tm
easu
res
4.5
on t
he R
icht
er s
cale
tha
n on
e th
at m
easu
res
3.5?
10 t
imes
1 & 2
1 % 41 & 3
3 & 22 & 3
25 & 75 & 77 & 5
Pra
ctic
e (A
vera
ge)
Pro
pert
ies
of L
ogar
ithm
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-3
10-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 10-3)
Rea
din
g t
o L
earn
Math
emati
csP
rope
rtie
s of
Log
arith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-3
10-3
©G
lenc
oe/M
cGra
w-H
ill58
9G
lenc
oe A
lgeb
ra 2
Lesson 10-3
Pre-
Act
ivit
yH
ow a
re t
he
pro
per
ties
of
exp
onen
ts a
nd
log
arit
hm
s re
late
d?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-3 a
t th
e to
p of
pag
e 54
1 in
you
r te
xtbo
ok.
Fin
d th
e va
lue
of lo
g 512
5.3
Fin
d th
e va
lue
of lo
g 55.
1F
ind
the
valu
e of
log 5
(125
)5)
.2
Whi
ch o
f th
e fo
llow
ing
stat
emen
ts is
tru
e?B
A.
log 5
(125
)5)
!(l
og5
125)
)(l
og5
5)B
.log
5(1
25 )
5) !
log 5
125
%lo
g 55
Rea
din
g t
he
Less
on
1.E
ach
of t
he p
rope
rtie
s of
loga
rith
ms
can
be s
tate
d in
wor
ds o
r in
sym
bols
.Com
plet
e th
est
atem
ents
of
thes
e pr
oper
ties
in w
ords
.
a.T
he lo
gari
thm
of
a qu
otie
nt is
the
of
the
loga
rith
ms
of t
he
and
the
.
b.T
he lo
gari
thm
of
a po
wer
is t
he
of t
he lo
gari
thm
of
the
base
and
the
.
c.T
he lo
gari
thm
of
a pr
oduc
t is
the
of
the
loga
rith
ms
of it
s
.
2.St
ate
whe
ther
eac
h of
the
fol
low
ing
equa
tion
s is
tru
eor
fal
se.I
f the
sta
tem
ent
is t
rue,
nam
e th
e pr
oper
ty o
f lo
gari
thm
s th
at is
illu
stra
ted.
a.lo
g 310
!lo
g 330
%lo
g 33
true
;Quo
tient
Pro
pert
yb.
log 4
12 !
log 4
4 (
log 4
8fa
lse
c.lo
g 281
!2
log 2
9tr
ue;P
ower
Pro
pert
yd.
log 8
30 !
log 8
5 '
log 8
6fa
lse
3.T
he a
lgeb
raic
pro
cess
of
solv
ing
the
equa
tion
log 2
x(
log 2
(x(
2) !
3 le
ads
to “
x!
%4
or x
!2.
”D
oes
this
mea
n th
at b
oth
%4
and
2 ar
e so
luti
ons
of t
he lo
gari
thm
ic e
quat
ion?
Exp
lain
you
r re
ason
ing.
Sam
ple
answ
er:N
o;2
is a
sol
utio
n be
caus
e it
chec
ks:l
og2
2 #
log 2
(2 #
2) !
log 2
2 #
log 2
4 !
1 #
2 !
3.H
owev
er,
beca
use
log 2
("4)
and
log 2
("2)
are
und
efin
ed,"
4 is
an
extr
aneo
usso
lutio
n an
d m
ust
be e
limin
ated
.The
onl
y so
lutio
n is
2.
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r so
met
hing
is t
o re
late
it s
omet
hing
you
alr
eady
kno
w.U
se w
ords
to e
xpla
in h
ow t
he P
rodu
ct P
rope
rty
for
expo
nent
s ca
n he
lp y
ou r
emem
ber
the
prod
uct
prop
erty
for
loga
rith
ms.
Sam
ple
answ
er:W
hen
you
mul
tiply
two
num
bers
or
expr
essi
ons
with
the
sam
e ba
se,y
ou a
ddth
e ex
pone
nts
and
keep
the
sam
e ba
se.L
ogar
ithm
s ar
e ex
pone
nts,
so t
o fin
d th
e lo
gari
thm
of
apr
oduc
t,yo
u ad
dth
e lo
gari
thm
s of
the
fac
tors
,kee
ping
the
sam
e ba
se.
fact
ors
sum
expo
nent
prod
uct
deno
min
ator
num
erat
ordi
ffer
ence
©G
lenc
oe/M
cGra
w-H
ill59
0G
lenc
oe A
lgeb
ra 2
Spi
rals
Con
side
r an
ang
le in
sta
ndar
d po
siti
on w
ith
its
vert
ex a
t a
poin
t O
calle
d th
epo
le.I
ts in
itia
l sid
e is
on
a co
ordi
nati
zed
axis
cal
led
the
pola
r ax
is.A
poi
nt P
on t
he t
erm
inal
sid
e of
the
ang
le is
nam
ed b
y th
e po
lar
coor
dina
tes
(r,.
),w
here
ris
the
dir
ecte
d di
stan
ce o
f th
e po
int
from
Oan
d .
is t
he m
easu
re o
fth
e an
gle.
Gra
phs
in t
his
syst
em m
ay b
e dr
awn
on p
olar
coo
rdin
ate
pape
rsu
ch a
s th
e ki
nd s
how
n be
low
.
1.U
se a
cal
cula
tor
to c
ompl
ete
the
tabl
e fo
r lo
g 2r
!& 12!
0&.
(Hin
t:To
fin
d !
on a
cal
cula
tor,
pres
s 12
0 r
2 .)
2.P
lot
the
poin
ts f
ound
in E
xerc
ise
1 on
the
gri
d ab
ove
and
conn
ect
to
form
a s
moo
th c
urve
.
Thi
s ty
pe o
f sp
iral
is c
alle
d a
loga
rith
mic
spi
ral b
ecau
se t
he a
ngle
m
easu
res
are
prop
orti
onal
to
the
loga
rith
ms
of t
he r
adii.
r1
23
45
67
8
!0+
120+
190+
240+
279+
310+
337+
360+
)
LOG
!)
LO
G"
01020
30
40
5060
7080
9010
011
012
013
0
140
150
160
170
180
190 200 21
0 220 23
024
025
026
027
028
029
030
031
0
32033
0340350
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-3
10-3
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
A
Answers (Lesson 10-4)
Stu
dy
Gu
ide
an
d I
nte
rven
tion
Com
mon
Log
arith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-4
10-4
©G
lenc
oe/M
cGra
w-H
ill59
1G
lenc
oe A
lgeb
ra 2
Lesson 10-4
Co
mm
on
Lo
gar
ith
ms
Bas
e 10
loga
rith
ms
are
calle
d co
mm
on l
ogar
ith
ms.
The
expr
essi
on lo
g 10
xis
usu
ally
wri
tten
wit
hout
the
sub
scri
pt a
s lo
g x.
Use
the
ke
y on
your
cal
cula
tor
to e
valu
ate
com
mon
loga
rith
ms.
The
rel
atio
n be
twee
n ex
pone
nts
and
loga
rith
ms
give
s th
e fo
llow
ing
iden
tity
.
Inve
rse
Pro
pert
y of
Log
arith
ms
and
Exp
onen
ts10
log
x!
x
Eva
luat
e lo
g 50
to
fou
r d
ecim
al p
lace
s.U
se t
he L
OG
key
on
your
cal
cula
tor.
To f
our
deci
mal
pla
ces,
log
50 !
1.69
90.
Sol
ve 3
2x#
1!
12.
32x
(1
!12
Orig
inal
equ
atio
n
log
32x
(1
!lo
g 12
Pro
pert
y of
Equ
ality
for
Loga
rithm
s
(2x
(1)
log
3 !
log
12P
ower
Pro
pert
y of
Log
arith
ms
2x(
1 !
Div
ide
each
sid
e by
log
3.
2x!
%1
Sub
trac
t 1 fr
om e
ach
side
.
x!
!%
1 "M
ultip
ly e
ach
side
by
.
x)
0.63
09
Use
a c
alcu
lato
r to
eva
luat
e ea
ch e
xpre
ssio
n t
o fo
ur
dec
imal
pla
ces.
1.lo
g 18
2.lo
g 39
3.lo
g 12
01.
2553
1.59
112.
0792
4.lo
g 5.
85.
log
42.3
6.lo
g 0.
003
0.76
341.
6263
"2.
5229
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.R
oun
d t
o fo
ur
dec
imal
pla
ces.
7.43
x!
120.
5975
8.6x
(2
!18
"0.
3869
9.54
x%
2!
120
1.24
3710
.73x
%1
,21
{x|x
)0.
8549
}
11.2
.4x
(4
!30
"0.
1150
12.6
.52x
,20
0{x
|x)
1.41
53}
13.3
.64x
%1
!85
.41.
1180
14.2
x(
5!
3x%
213
.966
6
15.9
3x!
45x
(2
"8.
1595
16.6
x%
5!
27x
(3
"3.
6069
1 & 2lo
g 12
& log
31 & 2lo
g 12
& log
3
log
12& lo
g 3
LOG
Exer
cises
Exer
cises
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
©G
lenc
oe/M
cGra
w-H
ill59
2G
lenc
oe A
lgeb
ra 2
Ch
ang
e o
f B
ase
Form
ula
The
fol
low
ing
form
ula
is u
sed
to c
hang
e ex
pres
sion
s w
ith
diff
eren
t lo
gari
thm
ic b
ases
to
com
mon
loga
rith
m e
xpre
ssio
ns.
Cha
nge
of B
ase
Form
ula
For
all
posi
tive
num
bers
a, b
, and
n, w
here
a"
1 an
d b
"1,
log a
n!
Exp
ress
log
815
in
ter
ms
of c
omm
on l
ogar
ith
ms.
Th
en a
pp
roxi
mat
eit
s va
lue
to f
our
dec
imal
pla
ces.
log 8
15!
Cha
nge
of B
ase
For
mul
a
)1.
3023
Sim
plify
.
The
val
ue o
f lo
g 815
is a
ppro
xim
atel
y 1.
3023
.
Exp
ress
eac
h l
ogar
ith
m i
n t
erm
s of
com
mon
log
arit
hm
s.T
hen
ap
pro
xim
ate
its
valu
e to
fou
r d
ecim
al p
lace
s.
1.lo
g 316
2.lo
g 240
3.lo
g 535
,2.5
237
,5.3
219
,2.2
091
4.lo
g 422
5.lo
g 12
200
6.lo
g 250
,2.2
297
,2.1
322
,5.6
439
7.lo
g 50.
48.
log 3
29.
log 4
28.5
,"0.
5693
,0.6
309
,2.4
164
10.l
og3
(20)
211
.log
6(5
)412
.log
8(4
)5
,5.4
537
,3.5
930
,3.3
333
13.l
og5
(8)3
14.l
og2
(3.6
)615
.log
12(1
0.5)
4
,3.8
761
,11.
0880
,3.7
851
16.l
og3
#15
0$
17.l
og4
3 #39$
18.l
og5
4 #16
00$
,2.2
804
,0.8
809
,1.1
460
log
1600
%%
4 lo
g 5
log
39% 3
log
4lo
g 15
0% 2
log
3
4 lo
g 10
.5%
%lo
g 12
6 lo
g 3.
6%
%lo
g 2
3 lo
g 8
%lo
g 5
5 lo
g 4
%lo
g 8
4 lo
g 5
%lo
g 6
2 lo
g 20
%%
log
3
log
28.5
%%
log
4lo
g 2
% log
3lo
g 0.
4% lo
g 5
log
50% lo
g 2
log
200
% log
12lo
g 22
% log
4
log
35% lo
g 5
log
40% lo
g 2
log
16% lo
g 3
log 10
15& lo
g 108
log b
n& lo
g ba
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Com
mon
Log
arith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-4
10-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A12 Glencoe Algebra 2
Answers (Lesson 10-4)
Skil
ls P
ract
ice
Com
mon
Log
arith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-4
10-4
©G
lenc
oe/M
cGra
w-H
ill59
3G
lenc
oe A
lgeb
ra 2
Lesson 10-4
Use
a c
alcu
lato
r to
eva
luat
e ea
ch e
xpre
ssio
n t
o fo
ur
dec
imal
pla
ces.
1.lo
g 6
0.77
822.
log
151.
1761
3.lo
g 1.
10.
0414
4.lo
g 0.
3"
0.52
29
Use
th
e fo
rmu
la p
H !
"lo
g[H
#]
to f
ind
th
e p
H o
f ea
ch s
ubs
tan
ce g
iven
its
con
cen
trat
ion
of
hyd
roge
n i
ons.
5.ga
stri
c ju
ices
:[H
(] !
1.0
/10
%1
mol
e pe
r lit
er1.
0
6.to
mat
o ju
ice:
[H(
] !7.
94 /
10%
5m
ole
per
liter
4.1
7.bl
ood:
[H(
] !3.
98 /
10%
8m
ole
per
liter
7.4
8.to
othp
aste
:[H
(] !
1.26
/10
%10
mol
e pe
r lit
er9.
9
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.R
oun
d t
o fo
ur
dec
imal
pla
ces.
9.3x
#24
3{x
|x$
5}10
.16v
+&v '
v(
"(
11.8
p!
501.
8813
12.7
y!
151.
3917
13.5
3b!
106
0.96
5914
.45k
!37
0.52
09
15.1
27p
!12
00.
2752
16.9
2m!
270.
75
17.3
r%
5!
4.1
6.28
4318
.8y
(4
#15
{y|y
$"
2.69
77}
19.7
.6d
(3
!57
.2"
1.00
4820
.0.5
t%
8!
16.3
3.97
32
21.4
2x2
!84
*1.
0888
22.5
x2(
1 !10
*0.
6563
Exp
ress
eac
h l
ogar
ith
m i
n t
erm
s of
com
mon
log
arit
hm
s.T
hen
ap
pro
xim
ate
its
valu
e to
fou
r d
ecim
al p
lace
s.
23.l
og3
7;1
.771
224
.log
566
;2.6
032
25.l
og2
35;5
.129
326
.log
610
;1.2
851
log 10
10%
%lo
g 106
log 10
35%
%lo
g 102
log 10
66%
%lo
g 105
log 10
7% lo
g 103
1 % 21 & 4
©G
lenc
oe/M
cGra
w-H
ill59
4G
lenc
oe A
lgeb
ra 2
Use
a c
alcu
lato
r to
eva
luat
e ea
ch e
xpre
ssio
n t
o fo
ur
dec
imal
pla
ces.
1.lo
g 10
12.
0043
2.lo
g 2.
20.
3424
3.lo
g 0.
05"
1.30
10
Use
th
e fo
rmu
la p
H !
"lo
g[H
#]
to f
ind
th
e p
H o
f ea
ch s
ubs
tan
ce g
iven
its
con
cen
trat
ion
of
hyd
roge
n i
ons.
4.m
ilk:[
H(
] !2.
51 /
10%
7m
ole
per
liter
6.6
5.ac
id r
ain:
[H(
] !2.
51 /
10%
6m
ole
per
liter
5.6
6.bl
ack
coff
ee:[
H(
] !1.
0 /
10%
5m
ole
per
liter
5.0
7.m
ilk o
f m
agne
sia:
[H(
] !3.
16 /
10%
11m
ole
per
liter
10.5
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.R
oun
d t
o fo
ur
dec
imal
pla
ces.
8.2x
$25
{x|x
'4.
6439
}9.
5a!
120
2.97
4610
.6z
!45
.62.
1319
11.9
m,
100
{m|m
)2.
0959
}12
.3.5
x!
47.9
3.08
8513
.8.2
y!
64.5
1.98
02
14.2
b(
1+
7.31
{b|b
(1.
8699
}15.
42x
!27
1.18
8716
.2a
%4
!82
.110
.359
3
17.9
z%
2#
38{z
|z$
3.65
55}
18.5
w(
3!
17"
1.23
9619
.30x
2!
50*
1.07
25
20.5
x2%
3!
72*
2.37
8521
.42x
!9x
(1
3.81
8822
.2n
(1
!52
n%
10.
9117
Exp
ress
eac
h l
ogar
ith
m i
n t
erm
s of
com
mon
log
arit
hm
s.T
hen
ap
pro
xim
ate
its
valu
e to
fou
r d
ecim
al p
lace
s.
23.l
og5
12;1
.544
024
.log
832
;1.6
667
25.l
og11
9 ;0
.916
3
26.l
og2
18
;4.1
699
27.l
og9
6;0
.815
528
.log
7#
8$;
29.H
ORT
ICU
LTU
RE
Sibe
rian
iris
es f
lour
ish
whe
n th
e co
ncen
trat
ion
of h
ydro
gen
ions
[H(
]in
the
soi
l is
not
less
tha
n 1.
58 /
10%
8m
ole
per
liter
.Wha
t is
the
pH
of
the
soil
in w
hich
thes
e ir
ises
will
flo
uris
h?7.
8 or
less
30.A
CID
ITY
The
pH
of
vine
gar
is 2
.9 a
nd t
he p
H o
f m
ilk is
6.6
.How
man
y ti
mes
gre
ater
isth
e hy
drog
en io
n co
ncen
trat
ion
of v
ineg
ar t
han
of m
ilk?
abou
t 50
00
31.B
IOLO
GY
The
re a
re in
itia
lly 1
000
bact
eria
in a
cul
ture
.The
num
ber
of b
acte
ria
doub
les
each
hou
r.T
he n
umbe
r of
bac
teri
a N
pres
ent
afte
r t
hour
s is
N!
1000
(2)t
.How
long
will
it t
ake
the
cult
ure
to in
crea
se t
o 50
,000
bac
teri
a?ab
out
5.6
h
32.S
OU
ND
An
equa
tion
for
loud
ness
Lin
dec
ibel
s is
giv
en b
y L
!10
log
R,w
here
Ris
the
soun
d’s
rela
tive
inte
nsit
y.A
n ai
r-ra
id s
iren
can
rea
ch 1
50 d
ecib
els
and
jet
engi
ne n
oise
can
reac
h 12
0 de
cibe
ls.H
ow m
any
tim
es g
reat
er is
the
rel
ativ
e in
tens
ity
of t
he a
ir-r
aid
sire
n th
an t
hat
of t
he je
t en
gine
noi
se?
1000
log 10
8% 2
log 10
7lo
g 106
%%
log 10
9lo
g 1018
%%
log 10
2
log 10
9%
%lo
g 1011
log 10
32%
%lo
g 108
log 10
12%
%lo
g 105
Pra
ctic
e (A
vera
ge)
Com
mon
Log
arith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-4
10-4
0.53
43
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
A
Answers (Lesson 10-4)
Rea
din
g t
o L
earn
Math
emati
csC
omm
on L
ogar
ithm
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-4
10-4
©G
lenc
oe/M
cGra
w-H
ill59
5G
lenc
oe A
lgeb
ra 2
Lesson 10-4
Pre-
Act
ivit
yW
hy
is a
log
arit
hm
ic s
cale
use
d t
o m
easu
re a
cid
ity?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-4 a
t th
e to
p of
pag
e 54
7 in
you
r te
xtbo
ok.
Whi
ch s
ubst
ance
is m
ore
acid
ic,m
ilk o
r to
mat
oes?
to
mat
oes
Rea
din
g t
he
Less
on
1.R
hond
a us
ed t
he f
ollo
win
g ke
ystr
okes
to
ente
r an
exp
ress
ion
on h
er g
raph
ing
calc
ulat
or:
17
The
cal
cula
tor
retu
rned
the
res
ult
1.23
0448
921.
Whi
ch o
f th
e fo
llow
ing
conc
lusi
ons
are
corr
ect?
a,c,
and
d
a.T
he b
ase
10 lo
gari
thm
of
17 is
abo
ut 1
.230
4.
b.T
he b
ase
17 lo
gari
thm
of
10 is
abo
ut 1
.230
4.
c.T
he c
omm
on lo
gari
thm
of
17 is
abo
ut 1
.230
449.
d.10
1.23
0448
921
is v
ery
clos
e to
17.
e.T
he c
omm
on lo
gari
thm
of
17 is
exa
ctly
1.2
3044
8921
.
2.M
atch
eac
h ex
pres
sion
fro
m t
he f
irst
col
umn
wit
h an
exp
ress
ion
from
the
sec
ond
colu
mn
that
has
the
sam
e va
lue.
a.lo
g 22
ivi.
log 4
1
b.lo
g 12
iii
ii.l
og2
8
c.lo
g 31
iii
i.lo
g 10
12
d.lo
g 5v
iv.l
og5
5
e.lo
g 10
00ii
v.lo
g 0.
1
3.C
alcu
lato
rs d
o no
t ha
ve k
eys
for
find
ing
base
8 lo
gari
thm
s di
rect
ly.H
owev
er,y
ou c
an u
se
a ca
lcul
ator
to
find
log 8
20 if
you
app
ly t
he
form
ula.
Whi
ch o
f th
e fo
llow
ing
expr
essi
ons
are
equa
l to
log 8
20?
B a
nd C
A.l
og20
8B
.C
.D
.
Hel
pin
g Y
ou
Rem
emb
er
4.So
met
imes
it is
eas
ier
to r
emem
ber
a fo
rmul
a if
you
can
sta
te it
in w
ords
.Sta
te t
hech
ange
of
base
for
mul
a in
wor
ds.
Sam
ple
answ
er:T
o ch
ange
the
loga
rith
m o
f a
num
ber
from
one
bas
e to
ano
ther
,div
ide
the
log
of t
he o
rigi
nal n
umbe
rin
the
old
bas
e by
the
log
of t
he n
ew b
ase
in t
he o
ld b
ase.
log
8& lo
g 20
log
20& lo
g 8
log 10
20& lo
g 108
chan
ge o
f ba
se
1 & 5
ENTE
R)
LO
G
©G
lenc
oe/M
cGra
w-H
ill59
6G
lenc
oe A
lgeb
ra 2
The
Slid
e R
ule
Bef
ore
the
inve
ntio
n of
ele
ctro
nic
calc
ulat
ors,
com
puta
tion
s w
ere
ofte
npe
rfor
med
on
a sl
ide
rule
.A s
lide
rule
is b
ased
on
the
idea
of
loga
rith
ms.
It h
astw
o m
ovab
le r
ods
labe
led
wit
h C
and
D s
cale
s.E
ach
of t
he s
cale
s is
loga
rith
mic
.
To m
ulti
ply
2 /
3 on
a s
lide
rule
,mov
e th
e C
rod
to
the
righ
t as
sho
wn
belo
w.Y
ou c
an f
ind
2 /
3 by
add
ing
log
2 to
log
3,an
d th
e sl
ide
rule
add
s th
ele
ngth
s fo
r yo
u.T
he d
ista
nce
you
get
is 0
.778
,or
the
loga
rith
m o
f 6.
Fol
low
th
e st
eps
to m
ake
a sl
ide
rule
.
1.U
se g
raph
pap
er t
hat
has
smal
l squ
ares
,suc
h as
10
squ
ares
to
the
inch
.Usi
ng t
he s
cale
s sh
own
at
the
righ
t,pl
ot t
he c
urve
y!
log
xfo
r x
!1,
1.5,
and
the
who
le n
umbe
rs f
rom
2 t
hrou
gh 1
0.M
ake
an o
bvio
us h
eavy
dot
for
eac
h po
int
plot
ted.
2.Yo
u w
ill n
eed
two
stri
ps o
f ca
rdbo
ard.
A
5-by
-7 in
dex
card
,cut
in h
alf
the
long
way
,w
ill w
ork
fine
.Tur
n th
e gr
aph
you
mad
e in
E
xerc
ise
1 si
dew
ays
and
use
it t
o m
ark
a lo
gari
thm
ic s
cale
on
each
of
the
two
stri
ps.T
he f
igur
e sh
ows
the
mar
k fo
r 2
bein
g dr
awn.
3.E
xpla
in h
ow t
o us
e a
slid
e ru
le t
o di
vide
8 b
y 2.
Line
up
the
2 on
th
e C
sca
le w
ith t
he 8
on
the
D s
cale
.The
quo
tient
is t
he
num
ber
on t
he D
sca
le b
elow
th
e 1
on t
he C
sca
le.
0
0.1
0.2
0.3
y
1 2
11.
52
y =
log
x
0.1
0.2
12
1 21
CD
2 4
3 6
45
67
89
83
57
9
log
6
log
3lo
g 2
12
34
56
78
9
12
34
56
78
9
C D
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-4
10-4
1–2.
See
st
uden
ts’w
ork.
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 10-5)
Stu
dy
Gu
ide
an
d I
nte
rven
tion
Bas
e e
and
Nat
ural
Log
arith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-5
10-5
©G
lenc
oe/M
cGra
w-H
ill59
7G
lenc
oe A
lgeb
ra 2
Lesson 10-5
Bas
e e
and
Nat
ura
l Lo
gar
ith
ms
The
irra
tion
al n
umbe
r e
)2.
7182
8… o
ften
occ
urs
as t
he b
ase
for
expo
nent
ial a
nd lo
gari
thm
ic f
unct
ions
tha
t de
scri
be r
eal-
wor
ld p
heno
men
a.
Nat
ural
Bas
e e
As
nin
crea
ses,
!1 (
"nap
proa
ches
e)
2.71
828…
.
ln x
!lo
g ex
The
fun
ctio
ns y
!ex
and
y!
ln x
are
inve
rse
func
tion
s.
Inve
rse
Pro
pert
y of
Bas
e e
and
Nat
ural
Log
arith
ms
eln
x!
xln
ex
!x
Nat
ural
bas
e ex
pres
sion
s ca
n be
eva
luat
ed u
sing
the
ex
and
ln k
eys
on y
our
calc
ulat
or.
Eva
luat
e ln
168
5.U
se a
cal
cula
tor.
ln 1
685
)7.
4295
Wri
te a
log
arit
hm
ic e
quat
ion
equ
ival
ent
to e
2x!
7.e2
x!
7 →
log e
7 !
2xor
2x
!ln
7
Eva
luat
e ln
e18
.U
se t
he I
nver
se P
rope
rty
of B
ase
ean
d N
atur
al L
ogar
ithm
s.ln
e18
!18
Use
a c
alcu
lato
r to
eva
luat
e ea
ch e
xpre
ssio
n t
o fo
ur
dec
imal
pla
ces.
1.ln
732
2.ln
84,
350
3.ln
0.7
354.
ln 1
006.
5958
11.3
427
"0.
3079
4.60
52
5.ln
0.0
824
6.ln
2.3
887.
ln 1
28,2
458.
ln 0
.006
14"
2.49
620.
8705
11.7
617
"5.
0929
Wri
te a
n e
quiv
alen
t ex
pon
enti
al o
r lo
gari
thm
ic e
quat
ion
.
9.e1
5!
x10
.e3x
!45
11.l
n 20
!x
12.l
n x
!8
ln x
!15
3x!
ln 4
5ex
!20
x!
e8
13.e
%5x
!0.
214
.ln
(4x)
!9.
615
.e8.
2!
10x
16.l
n 0.
0002
!x
"5x
!ln
0.2
4x!
e9.6
ln 1
0x!
8.2
ex!
0.00
02
Eva
luat
e ea
ch e
xpre
ssio
n.
17.l
n e3
18.e
ln 4
219
.eln
0.5
20.l
n e1
6.2
342
0.5
16.2
1 & n
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exam
ple3
Exam
ple3
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill59
8G
lenc
oe A
lgeb
ra 2
Equ
atio
ns
and
Ineq
ual
itie
s w
ith
ean
d ln
All
prop
erti
es o
f lo
gari
thm
s fr
omea
rlie
r le
sson
s ca
n be
use
d to
sol
ve e
quat
ions
and
ineq
ualit
ies
wit
h na
tura
l log
arit
hms.
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.
a.3e
2x(
2 !
103e
2x(
2 !
10O
rigin
al e
quat
ion
3e2x
!8
Sub
trac
t 2 fr
om e
ach
side
.
e2x
!D
ivid
e ea
ch s
ide
by 3
.
ln e
2x!
ln
Pro
pert
y of
Equ
ality
for
Loga
rithm
s
2x!
ln
Inve
rse
Pro
pert
y of
Exp
onen
ts a
nd L
ogar
ithm
s
x!
ln
Mul
tiply
eac
h si
de b
y &1 2& .
x)
0.49
04U
se a
cal
cula
tor.
b.ln
(4x
%1)
$2
ln (
4x%
1) $
2O
rigin
al in
equa
lity
eln
(4x
%1)
$e2
Writ
e ea
ch s
ide
usin
g ex
pone
nts
and
base
e.
0 $
4x%
1 $
e2In
vers
e P
rope
rty
of E
xpon
ents
and
Log
arith
ms
1 $
4x$
e2(
1A
dditi
on P
rope
rty
of In
equa
litie
s
$x
$(e
2(
1)M
ultip
licat
ion
Pro
pert
y of
Ineq
ualit
ies
0.25
$x
$2.
0973
Use
a c
alcu
lato
r.
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.
1.e4
x!
120
2.ex
+25
3.ex
%2
(4
!21
1.19
69{x
|x(
3.21
89}
4.83
32
4.ln
6x
,4
5.ln
(x
(3)
%5
!%
26.
e%8x
+50
x)
9.09
9717
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5{x
|x)
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4890
}
7.e4
x%
1%
3 !
128.
ln (
5x(
3) !
3.6
9.2e
3x(
5 !
20.
9270
6.71
96no
sol
utio
n
10.6
(3e
x(
1!
2111
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(2x
%5)
!8
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n 5x
(ln
3x
#9
0.60
9414
92.9
790
{x|x
$23
.242
3}
1 & 41 & 4
8 & 31 & 2
8 & 38 & 3
8 & 3
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Bas
e e
and
Nat
ural
Log
arith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-5
10-5
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A15 Glencoe Algebra 2
A
Answers (Lesson 10-5)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-5
10-5
©G
lenc
oe/M
cGra
w-H
ill59
9G
lenc
oe A
lgeb
ra 2
Lesson 10-5
Use
a c
alcu
lato
r to
eva
luat
e ea
ch e
xpre
ssio
n t
o fo
ur
dec
imal
pla
ces.
1.e3
20.0
855
2.e%
20.
1353
3.ln
20.
6931
4.ln
0.0
9"
2.40
79
Wri
te a
n e
quiv
alen
t ex
pon
enti
al o
r lo
gari
thm
ic e
quat
ion
.
5.ex
!3
x!
ln 3
6.e4
!8x
4 !
ln 8
x
7.ln
15
!x
ex!
158.
ln x
)0.
6931
x)
e0.6
931
Eva
luat
e ea
ch e
xpre
ssio
n.
9.el
n 3
310
.eln
2x
2x
11.l
n e%
2.5
"2.
512
.ln
eyy
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.
13.e
x,
5{x
|x)
1.60
94}
14.e
x$
3.2
{x|x
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1632
}
15.2
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1 !
111.
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181.
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301.
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0.57
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n (x
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9.38
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3) !
1"
0.28
17
25.l
n (x
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n x
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1.92
21
Skil
ls P
ract
ice
Bas
e e
and
Nat
ural
Log
arith
ms
©G
lenc
oe/M
cGra
w-H
ill60
0G
lenc
oe A
lgeb
ra 2
Use
a c
alcu
lato
r to
eva
luat
e ea
ch e
xpre
ssio
n t
o fo
ur
dec
imal
pla
ces.
1.e1
.54.
4817
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82.
0794
3.ln
3.2
1.16
324.
e%0.
60.
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37"
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68
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te a
n e
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alen
t ex
pon
enti
al o
r lo
gari
thm
ic e
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ion
.
9.ln
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!x
10.l
n 36
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11.l
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n 9.
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9.3
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x15
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x!
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5 !
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ln 4
2 !
ln (
x#
1)
Eva
luat
e ea
ch e
xpre
ssio
n.
17.e
ln 1
212
18.e
ln 3
x3x
19.l
n e%
1"
120
.ln
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ve e
ach
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atio
n o
r in
equ
alit
y.
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INV
ESTI
NG
For
Exe
rcis
es 4
5 an
d 4
6,u
se t
he
form
ula
for
con
tin
uou
sly
com
pou
nd
ed i
nte
rest
,A!
Per
t ,w
her
e P
is t
he
pri
nci
pal
,ris
th
e an
nu
al i
nte
rest
rate
,an
d t
is t
he
tim
e in
yea
rs.
45.I
f Sa
rita
dep
osit
s $1
000
in a
n ac
coun
t pa
ying
3.4
% a
nnua
l int
eres
t co
mpo
unde
dco
ntin
uous
ly,w
hat
is t
he b
alan
ce in
the
acc
ount
aft
er 5
yea
rs?
$118
5.30
46.H
ow lo
ng w
ill it
tak
e th
e ba
lanc
e in
Sar
ita’
s ac
coun
t to
rea
ch $
2000
?ab
out
20.4
yr
47.R
AD
IOA
CTI
VE
DEC
AY
The
am
ount
of
a ra
dioa
ctiv
e su
bsta
nce
yth
at r
emai
ns a
fter
t
year
s is
giv
en b
y th
e eq
uati
on y
!ae
kt,w
here
ais
the
init
ial a
mou
nt p
rese
nt a
nd k
isth
e de
cay
cons
tant
for
the
rad
ioac
tive
sub
stan
ce.I
f a!
100,
y!
50,a
nd k
!%
0.03
5,fi
nd t
.ab
out
19.8
yr
Pra
ctic
e (A
vera
ge)
Bas
e e
and
Nat
ural
Log
arith
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-5
10-5
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 10-5)
Rea
din
g t
o L
earn
Math
emati
csB
ase
ean
d N
atur
al L
ogar
ithm
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-5
10-5
©G
lenc
oe/M
cGra
w-H
ill60
1G
lenc
oe A
lgeb
ra 2
Lesson 10-5
Pre-
Act
ivit
yH
ow i
s th
e n
atu
ral
base
eu
sed
in
ban
kin
g?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-5 a
t th
e to
p of
pag
e 55
4 in
you
r te
xtbo
ok.
Supp
ose
that
you
dep
osit
$67
5 in
a s
avin
gs a
ccou
nt t
hat
pays
an
annu
alin
tere
st r
ate
of 5
%.I
n ea
ch c
ase
liste
d be
low
,ind
icat
e w
hich
met
hod
ofco
mpo
undi
ng w
ould
res
ult
in m
ore
mon
ey in
you
r ac
coun
t at
the
end
of
one
year
.a.
annu
al c
ompo
undi
ng o
r m
onth
ly c
ompo
undi
ngm
onth
lyb.
quar
terl
y co
mpo
undi
ng o
r da
ily c
ompo
undi
ngda
ilyc.
daily
com
poun
ding
or
cont
inuo
us c
ompo
undi
ngco
ntin
uous
Rea
din
g t
he
Less
on
1.Ja
gdis
h en
tere
d th
e fo
llow
ing
keys
trok
es in
his
cal
cula
tor:
5
The
cal
cula
tor
retu
rned
the
res
ult
1.60
9437
912.
Whi
ch o
f th
e fo
llow
ing
conc
lusi
ons
are
corr
ect?
d an
d f
a.T
he c
omm
on lo
gari
thm
of
5 is
abo
ut 1
.609
4.
b.T
he n
atur
al lo
gari
thm
of
5 is
exa
ctly
1.6
0943
7912
.
c.T
he b
ase
5 lo
gari
thm
of e
is a
bout
1.6
094.
d.T
he n
atur
al lo
gari
thm
of
5 is
abo
ut 1
.609
438.
e.10
1.60
9437
912
is v
ery
clos
e to
5.
f.e1
.609
4379
12is
ver
y cl
ose
to 5
.
2.M
atch
eac
h ex
pres
sion
fro
m t
he f
irst
col
umn
wit
h it
s va
lue
in t
he s
econ
d co
lum
n.So
me
choi
ces
may
be
used
mor
e th
an o
nce
or n
ot a
t al
l.
a.el
n 5
IVI.
1
b.ln
1V
II.1
0
c.el
n e
VI
III.
%1
d.ln
e5
IVIV
.5
e.ln
eI
V.0
f.ln
!"I
IIV
I.e
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
som
ethi
ng is
to
expl
ain
it t
o so
meo
ne e
lse.
Supp
ose
that
you
are
stud
ying
wit
h a
clas
smat
e w
ho is
puz
zled
whe
n as
ked
to e
valu
ate
ln e
3 .H
ow w
ould
you
expl
ain
to h
im a
n ea
sy w
ay t
o fi
gure
thi
s ou
t?S
ampl
e an
swer
:ln
mea
ns n
atur
allo
g.Th
e na
tura
l log
of
e3is
the
pow
er t
o w
hich
you
rai
se e
to g
et e
3 .Th
isis
obv
ious
ly 3
.
1 & e
ENTE
R)
LN
©G
lenc
oe/M
cGra
w-H
ill60
2G
lenc
oe A
lgeb
ra 2
App
roxi
mat
ions
for
"an
d e
The
fol
low
ing
expr
essi
on c
an b
e us
ed t
o ap
prox
imat
e e.
If g
reat
er a
nd g
reat
erva
lues
of n
are
used
,the
val
ue o
f th
e ex
pres
sion
app
roxi
mat
es e
mor
e an
dm
ore
clos
ely.
!1 (
& n1 & "n
Ano
ther
way
to
appr
oxim
ate
eis
to
use
this
infin
ite
sum
.The
gre
ater
the
valu
e of
n,t
he c
lose
r th
e ap
prox
imat
ion.
e!
1 (
1 (
&1 2&(
& 21 '
3&(
& 2'
1 3'
4&
(…
(& 2
'3
'41 '
…'
n&
(…
In a
sim
ilar
man
ner,
*ca
n be
app
roxi
mat
ed u
sing
an
infin
ite
prod
uct
disc
over
ed b
y th
e E
nglis
h m
athe
mat
icia
n Jo
hn W
allis
(16
16–1
703)
.
&* 2&!
&2 1&'
&2 3&'
&4 3&'
&4 5&'
&6 5&'
&6 7&'
… '
& 2n2 %n
1&
' & 2n
2 (n1
&…
Sol
ve e
ach
pro
blem
.
1.U
se a
cal
cula
tor
wit
h an
ex
key
to f
ind
eto
7 d
ecim
al p
lace
s.2.
7182
818
2.U
se t
he e
xpre
ssio
n !1
(& n1 & "n
to a
ppro
xim
ate
eto
3 d
ecim
al p
lace
s.U
se
5,10
0,50
0,an
d 70
00 a
s va
lues
of n
.2.
488,
2.70
5,2.
716,
2.71
8
3.U
se t
he in
finit
e su
m t
o ap
prox
imat
e e
to 3
dec
imal
pla
ces.
Use
the
who
le
num
bers
fro
m 3
thr
ough
6 a
s va
lues
of n
.2.
667,
2.70
8,2.
717,
2.71
8
4.W
hich
app
roxi
mat
ion
met
hod
appr
oach
es t
he v
alue
of e
mor
e qu
ickl
y?th
e in
finite
sum
5.U
se a
cal
cula
tor
wit
h a
*ke
y to
find
*to
7 d
ecim
al p
lace
s.3.
1415
927
6.U
se t
he in
finit
e pr
oduc
t to
app
roxi
mat
e *
to 3
dec
imal
pla
ces.
Use
the
w
hole
num
bers
fro
m 3
thr
ough
6 a
s va
lues
of n
.2.
926,
2.97
2,3.
002,
3.02
3
7.D
oes
the
infin
ite
prod
uct
give
goo
d ap
prox
imat
ions
for
*qu
ickl
y?no
8.Sh
ow t
hat
*4
(*
5is
equ
al t
o e6
to 4
dec
imal
pla
ces.
To 4
dec
imal
pla
ces,
they
bot
h eq
ual 4
03.4
288.
9.W
hich
is la
rger
,e*
or *
e ?e&
> &
e
10.T
he e
xpre
ssio
n x
reac
hes
a m
axim
um v
alue
at
x!
e.U
se t
his
fact
to
prov
e th
e in
equa
lity
you
foun
d in
Exe
rcis
e 9.
e%1 e%>
&% &1 % ;#
e%1 e% $&e
> # &
% &1 % $&e ;e
&>
&e
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-5
10-5
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
A
Answers (Lesson 10-6)
Stu
dy
Gu
ide
an
d I
nte
rven
tion
Exp
onen
tial G
row
th a
nd D
ecay
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-6
10-6
©G
lenc
oe/M
cGra
w-H
ill60
3G
lenc
oe A
lgeb
ra 2
Lesson 10-6
Exp
on
enti
al D
ecay
Dep
reci
atio
n of
val
ue a
nd r
adio
acti
ve d
ecay
are
exa
mpl
es o
fex
pon
enti
al d
ecay
.Whe
n a
quan
tity
dec
reas
es b
y a
fixe
d pe
rcen
t ea
ch t
ime
peri
od,t
heam
ount
of
the
quan
tity
aft
er t
tim
e pe
riod
s is
giv
en b
y y
!a(
1 %
r)t ,
whe
re a
is t
he in
itia
lam
ount
and
ris
the
per
cent
dec
reas
e ex
pres
sed
as a
dec
imal
.A
noth
er e
xpon
enti
al d
ecay
mod
el o
ften
use
d by
sci
enti
sts
is y
!ae
%kt
,whe
re k
is a
con
stan
t.
CO
NSU
MER
PR
ICES
As
tech
nol
ogy
adva
nce
s,th
e p
rice
of
man
yte
chn
olog
ical
dev
ices
su
ch a
s sc
ien
tifi
c ca
lcu
lato
rs a
nd
cam
cord
ers
goes
dow
n.
On
e br
and
of
han
d-h
eld
org
aniz
er s
ells
for
$89
.
a.If
its
pri
ce d
ecre
ases
by
6% p
er y
ear,
how
mu
ch w
ill
it c
ost
afte
r 5
year
s?U
se t
he e
xpon
enti
al d
ecay
mod
el w
ith
init
ial a
mou
nt $
89,p
erce
nt d
ecre
ase
0.06
,and
tim
e 5
year
s.y
!a(
1 %
r)t
Exp
onen
tial d
ecay
form
ula
y!
89(1
%0.
06)5
a!
89, r
!0.
06, t
!5
y!
$65.
32A
fter
5 y
ears
the
pri
ce w
ill b
e $6
5.32
.
b.A
fter
how
man
y ye
ars
wil
l it
s p
rice
be
$50?
To fi
nd w
hen
the
pric
e w
ill b
e $5
0,ag
ain
use
the
expo
nent
ial d
ecay
form
ula
and
solv
e fo
r t.
y!
a(1
%r)
tE
xpon
entia
l dec
ay fo
rmul
a
50 !
89(1
%0.
06)t
y!
50, a
!89
, r!
0.06
!(0
.94)
tD
ivid
e ea
ch s
ide
by 8
9.
log
!"!
log
(0.9
4)t
Pro
pert
y of
Equ
ality
for
Loga
rithm
s
log
!"!
tlo
g 0.
94P
ower
Pro
pert
y
t!
Div
ide
each
sid
e by
log
0.94
.
t)
9.3
The
pri
ce w
ill b
e $5
0 af
ter
abou
t 9.
3 ye
ars.
1.B
USI
NES
SA
fur
nitu
re s
tore
is c
losi
ng o
ut it
s bu
sine
ss.E
ach
wee
k th
e ow
ner
low
ers
pric
es b
y 25
%.A
fter
how
man
y w
eeks
will
the
sal
e pr
ice
of a
$50
0 it
em d
rop
belo
w $
100?
6 w
eeks
CA
RB
ON
DA
TIN
GU
se t
he
form
ula
y!
ae"
0.00
012t
,wh
ere
ais
th
e in
itia
l am
oun
t of
Car
bon
-14,
tis
th
e n
um
ber
of y
ears
ago
th
e an
imal
liv
ed,a
nd
yis
th
e re
mai
nin
gam
oun
t af
ter
tye
ars.
2.H
ow o
ld is
a fo
ssil
rem
ain
that
has
lost
95%
of i
ts C
arbo
n-14
?ab
out 2
5,00
0 ye
ars
old
3.H
ow o
ld is
a s
kele
ton
that
has
95%
of i
ts C
arbo
n-14
rem
aini
ng?
abou
t 427
.5 y
ears
old
log
!&5 80 9&"
&&
log
0.94
50 & 8950 & 8950 & 89
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill60
4G
lenc
oe A
lgeb
ra 2
Exp
on
enti
al G
row
thPo
pula
tion
incr
ease
and
gro
wth
of
bact
eria
col
onie
s ar
e ex
ampl
esof
exp
onen
tial
gro
wth
.Whe
n a
quan
tity
incr
ease
s by
a fi
xed
perc
ent
each
tim
e pe
riod
,the
amou
nt o
f th
at q
uant
ity
afte
r t
tim
e pe
riod
s is
giv
en b
y y
!a(
1 (
r)t ,
whe
re a
is t
he in
itia
lam
ount
and
ris
the
per
cent
incr
ease
(or
rat
e of
gro
wth
) ex
pres
sed
as a
dec
imal
.A
noth
er e
xpon
enti
al g
row
th m
odel
oft
en u
sed
by s
cien
tist
s is
y!
aekt
,whe
re k
is a
con
stan
t.
A c
omp
ute
r en
gin
eer
is h
ired
for
a s
alar
y of
$28
,000
.If
she
gets
a5%
rai
se e
ach
yea
r,af
ter
how
man
y ye
ars
wil
l sh
e be
mak
ing
$50,
000
or m
ore?
Use
the
exp
onen
tial
gro
wth
mod
el w
ith
a!
28,0
00,y
!50
,000
,and
r!
0.05
and
sol
ve f
or t
.
y!
a(1
(r)
tE
xpon
entia
l gro
wth
form
ula
50,0
00 !
28,0
00(1
(0.
05)t
y!
50,0
00, a
!28
,000
, r!
0.05
!(1
.05)
tD
ivid
e ea
ch s
ide
by 2
8,00
0.
log
!"!
log
(1.0
5)t
Pro
pert
y of
Equ
ality
of L
ogar
ithm
s
log
!"!
tlo
g 1.
05P
ower
Pro
pert
y
t!
Div
ide
each
sid
e by
log
1.05
.
t)
11.9
yea
rsU
se a
cal
cula
tor.
If r
aise
s ar
e gi
ven
annu
ally
,she
will
be
mak
ing
over
$50
,000
in 1
2 ye
ars.
1.B
AC
TER
IA G
RO
WTH
A c
erta
in s
trai
n of
bac
teri
a gr
ows
from
40
to 3
26 in
120
min
utes
.F
ind
kfo
r th
e gr
owth
for
mul
a y
!ae
kt,w
here
tis
in m
inut
es.
abou
t 0.
0175
2.IN
VES
TMEN
TC
arl p
lans
to
inve
st $
500
at 8
.25%
inte
rest
,com
poun
ded
cont
inuo
usly
.H
ow lo
ng w
ill it
tak
e fo
r hi
s m
oney
to
trip
le?
abou
t 14
yea
rs
3.SC
HO
OL
POPU
LATI
ON
The
re a
re c
urre
ntly
850
stu
dent
s at
the
hig
h sc
hool
,whi
chre
pres
ents
ful
l cap
acit
y.T
he t
own
plan
s an
add
itio
n to
hou
se 4
00 m
ore
stud
ents
.If
the
scho
ol p
opul
atio
n gr
ows
at 7
.8%
per
yea
r,in
how
man
y ye
ars
will
the
new
add
itio
nbe
ful
l?ab
out
5 ye
ars
4.EX
ERC
ISE
Hug
o be
gins
a w
alki
ng p
rogr
am b
y w
alki
ng
mile
per
day
for
one
wee
k.
Eac
h w
eek
ther
eaft
er h
e in
crea
ses
his
mile
age
by 1
0%.A
fter
how
man
y w
eeks
is h
ew
alki
ng m
ore
than
5 m
iles
per
day?
24 w
eeks
5.V
OC
AB
ULA
RY G
RO
WTH
Whe
n E
mily
was
18
mon
ths
old,
she
had
a 10
-wor
dvo
cabu
lary
.By
the
tim
e sh
e w
as 5
yea
rs o
ld (6
0 m
onth
s),h
er v
ocab
ular
y w
as 2
500
wor
ds.
If h
er v
ocab
ular
y in
crea
sed
at a
con
stan
t pe
rcen
t pe
r m
onth
,wha
t w
as t
hat
incr
ease
?ab
out 1
4%
1 & 2
log
!&5 20 8&"
& log
1.05
50 & 2850 & 2850 & 28
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Exp
onen
tial G
row
th a
nd D
ecay
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-6
10-6
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A18 Glencoe Algebra 2
Answers (Lesson 10-6)
Skil
ls P
ract
ice
Exp
onen
tial G
row
th a
nd D
ecay
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-6
10-6
©G
lenc
oe/M
cGra
w-H
ill60
5G
lenc
oe A
lgeb
ra 2
Lesson 10-6
Sol
ve e
ach
pro
blem
.
1.FI
SHIN
GIn
an
over
-fis
hed
area
,the
cat
ch o
f a
cert
ain
fish
is d
ecre
asin
g at
an
aver
age
rate
of
8% p
er y
ear.
If t
his
decl
ine
pers
ists
,how
long
will
it t
ake
for
the
catc
h to
rea
chha
lf o
f th
e am
ount
bef
ore
the
decl
ine?
abou
t 8.
3 yr
2.IN
VES
TIN
GA
lex
inve
sts
$200
0 in
an
acco
unt
that
has
a 6
% a
nnua
l rat
e of
gro
wth
.To
the
near
est
year
,whe
n w
ill t
he in
vest
men
t be
wor
th $
3600
?10
yr
3.PO
PULA
TIO
NA
cur
rent
cen
sus
show
s th
at t
he p
opul
atio
n of
a c
ity
is 3
.5 m
illio
n.U
sing
the
form
ula
P!
aert
,fin
d th
e ex
pect
ed p
opul
atio
n of
the
cit
y in
30
year
s if
the
gro
wth
rate
rof
the
pop
ulat
ion
is 1
.5%
per
yea
r,a
repr
esen
ts t
he c
urre
nt p
opul
atio
n in
mill
ions
,an
d t
repr
esen
ts t
he t
ime
in y
ears
.ab
out
5.5
mill
ion
4.PO
PULA
TIO
NT
he p
opul
atio
n P
in t
hous
ands
of
a ci
ty c
an b
e m
odel
ed b
y th
e eq
uati
onP
!80
e0.0
15t ,
whe
re t
is t
he t
ime
in y
ears
.In
how
man
y ye
ars
will
the
pop
ulat
ion
of t
heci
ty b
e 12
0,00
0?ab
out
27 y
r
5.B
AC
TER
IAH
ow m
any
days
will
it t
ake
a cu
ltur
e of
bac
teri
a to
incr
ease
fro
m 2
000
to50
,000
if t
he g
row
th r
ate
per
day
is 9
3.2%
?ab
out
4.9
days
6.N
UC
LEA
R P
OW
ERT
he e
lem
ent
plut
oniu
m-2
39 is
hig
hly
radi
oact
ive.
Nuc
lear
rea
ctor
sca
n pr
oduc
e an
d al
so u
se t
his
elem
ent.
The
hea
t th
at p
luto
nium
-239
em
its
has
help
ed t
opo
wer
equ
ipm
ent
on t
he m
oon.
If t
he h
alf-
life
of p
luto
nium
-239
is 2
4,36
0 ye
ars,
wha
t is
the
valu
e of
kfo
r th
is e
lem
ent?
abou
t 0.
0000
2845
7.D
EPR
ECIA
TIO
NA
Glo
bal P
osit
ioni
ng S
atel
lite
(GP
S) s
yste
m u
ses
sate
llite
info
rmat
ion
to lo
cate
gro
und
posi
tion
.Abu
’s s
urve
ying
fir
m b
ough
t a
GP
S sy
stem
for
$12
,500
.The
GP
S de
prec
iate
d by
a f
ixed
rat
e of
6%
and
is n
ow w
orth
$86
00.H
ow lo
ng a
go d
id A
bubu
y th
e G
PS
syst
em?
abou
t 6.
0 yr
8.B
IOLO
GY
In a
labo
rato
ry,a
n or
gani
sm g
row
s fr
om 1
00 t
o 25
0 in
8 h
ours
.Wha
t is
the
hour
ly g
row
th r
ate
in t
he g
row
th f
orm
ula
y!
a(1
(r)
t ?ab
out
12.1
3%
©G
lenc
oe/M
cGra
w-H
ill60
6G
lenc
oe A
lgeb
ra 2
Sol
ve e
ach
pro
blem
.
1.IN
VES
TIN
GT
he fo
rmul
a A
!P!1
("2t
give
s th
e va
lue
of a
n in
vest
men
t af
ter
tye
ars
in
an a
ccou
nt t
hat
earn
s an
ann
ual i
nter
est
rate
rco
mpo
unde
d tw
ice
a ye
ar.S
uppo
se $
500
is in
vest
ed a
t 6%
ann
ual i
nter
est
com
poun
ded
twic
e a
year
.In
how
man
y ye
ars
will
the
inve
stm
ent
be w
orth
$10
00?
abou
t 11
.7 y
r
2.B
AC
TER
IAH
ow m
any
hour
s w
ill it
tak
e a
cult
ure
of b
acte
ria
to in
crea
se f
rom
20
to20
00 if
the
gro
wth
rat
e pe
r ho
ur is
85%
?ab
out
7.5
h
3.R
AD
IOA
CTI
VE
DEC
AY
A r
adio
acti
ve s
ubst
ance
has
a h
alf-
life
of 3
2 ye
ars.
Fin
d th
eco
nsta
nt k
in t
he d
ecay
for
mul
a fo
r th
e su
bsta
nce.
abou
t 0.
0216
6
4.D
EPR
ECIA
TIO
NA
pie
ce o
f m
achi
nery
val
ued
at $
250,
000
depr
ecia
tes
at a
fix
ed r
ate
of12
% p
er y
ear.
Aft
er h
ow m
any
year
s w
ill t
he v
alue
hav
e de
prec
iate
d to
$10
0,00
0?ab
out
7.2
yr
5.IN
FLA
TIO
NFo
r D
ave
to b
uy a
new
car
com
para
bly
equi
pped
to
the
one
he b
ough
t 8
year
sag
o w
ould
cos
t $1
2,50
0.Si
nce
Dav
e bo
ught
the
car
,the
infl
atio
n ra
te f
or c
ars
like
his
has
been
at
an a
vera
ge a
nnua
l rat
e of
5.1
%.I
f D
ave
orig
inal
ly p
aid
$840
0 fo
r th
e ca
r,ho
wlo
ng a
go d
id h
e bu
y it
?ab
out
8 yr
6.R
AD
IOA
CTI
VE
DEC
AY
Cob
alt,
an e
lem
ent
used
to
mak
e al
loys
,has
sev
eral
isot
opes
.O
ne o
f th
ese,
coba
lt-6
0,is
rad
ioac
tive
and
has
a h
alf-
life
of 5
.7 y
ears
.Cob
alt-
60 is
use
d to
trac
e th
e pa
th o
f no
nrad
ioac
tive
sub
stan
ces
in a
sys
tem
.Wha
t is
the
val
ue o
f kfo
rC
obal
t-60
?ab
out
0.12
16
7.W
HA
LES
Mod
ern
wha
les
appe
ared
5%
10 m
illio
n ye
ars
ago.
The
ver
tebr
ae o
f a
wha
ledi
scov
ered
by
pale
onto
logi
sts
cont
ain
roug
hly
0.25
% a
s m
uch
carb
on-1
4 as
the
y w
ould
have
con
tain
ed w
hen
the
wha
le w
as a
live.
How
long
ago
did
the
wha
le d
ie?
Use
k
!0.
0001
2.ab
out
50,0
00 y
r
8.PO
PULA
TIO
NT
he p
opul
atio
n of
rab
bits
in a
n ar
ea is
mod
eled
by
the
grow
th e
quat
ion
P(t
) !8e
0.26
t ,w
here
Pis
in t
hous
ands
and
tis
in y
ears
.How
long
will
it t
ake
for
the
popu
lati
on t
o re
ach
25,0
00?
abou
t 4.
4 yr
9.D
EPR
ECIA
TIO
NA
com
pute
r sy
stem
dep
reci
ates
at
an a
vera
ge r
ate
of 4
% p
er m
onth
.If
the
valu
e of
the
com
pute
r sy
stem
was
ori
gina
lly $
12,0
00,i
n ho
w m
any
mon
ths
is it
wor
th $
7350
?ab
out
12 m
o
10.B
IOLO
GY
In a
labo
rato
ry,a
cul
ture
incr
ease
s fr
om 3
0 to
195
org
anis
ms
in 5
hou
rs.
Wha
t is
the
hou
rly
grow
th r
ate
in t
he g
row
th f
orm
ula
y!
a(1
(r)
t ?ab
out
45.4
%
r & 2
Pra
ctic
e (A
vera
ge)
Exp
onen
tial G
row
th a
nd D
ecay
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-6
10-6
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
A
Answers (Lesson 10-6)
Rea
din
g t
o L
earn
Math
emati
csE
xpon
entia
l Gro
wth
and
Dec
ay
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-6
10-6
©G
lenc
oe/M
cGra
w-H
ill60
7G
lenc
oe A
lgeb
ra 2
Lesson 10-6
Pre-
Act
ivit
yH
ow c
an y
ou d
eter
min
e th
e cu
rren
t va
lue
of y
our
car?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-6 a
t th
e to
p of
pag
e 56
0 in
you
r te
xtbo
ok.
•B
etw
een
whi
ch t
wo
year
s sh
own
in t
he t
able
did
the
car
dep
reci
ate
byth
e gr
eate
st a
mou
nt?
betw
een
year
s 0
and
1•
Des
crib
e tw
o w
ays
to c
alcu
late
the
val
ue o
f th
e ca
r 6
year
s af
ter
it w
aspu
rcha
sed.
(Do
not
actu
ally
cal
cula
te t
he v
alue
.)S
ampl
e an
swer
:1.M
ultip
ly $
9200
.66
by 0
.16
and
subt
ract
the
resu
lt fr
om $
9200
.66.
2.M
ultip
ly $
9200
.66
by 0
.84.
Rea
din
g t
he
Less
on
1.St
ate
whe
ther
eac
h si
tuat
ion
is a
n ex
ampl
e of
exp
onen
tial
gro
wth
or d
ecay
.
a.A
cit
y ha
d 42
,000
res
iden
ts in
198
0 an
d 12
8,00
0 re
side
nts
in 2
000.
grow
th
b.R
aul c
ompa
red
the
valu
e of
his
car
whe
n he
bou
ght
it n
ew t
o th
e va
lue
whe
n he
trad
ed ‘;
lpit
in s
ix y
ears
late
r.de
cay
c.A
pal
eont
olog
ist
com
pare
d th
e am
ount
of
carb
on-1
4 in
the
ske
leto
n of
an
anim
alw
hen
it d
ied
to t
he a
mou
nt 3
00 y
ears
late
r.de
cay
d.M
aria
dep
osit
ed $
750
in a
sav
ings
acc
ount
pay
ing
4.5%
ann
ual i
nter
est
com
poun
ded
quar
terl
y.Sh
e di
d no
t m
ake
any
wit
hdra
wal
s or
fur
ther
dep
osit
s.Sh
e co
mpa
red
the
bala
nce
in h
er p
assb
ook
imm
edia
tely
aft
er s
he o
pene
d th
e ac
coun
t to
the
bal
ance
3
year
s la
ter.
grow
th
2.St
ate
whe
ther
eac
h eq
uati
on r
epre
sent
s ex
pone
ntia
l gro
wth
or
deca
y.
a.y
!5e
0.15
tgr
owth
b.y
!10
00(1
%0.
05)t
deca
y
c.y
!0.
3e%
1200
tde
cay
d.y
!2(
1 (
0.00
01)t
grow
th
Hel
pin
g Y
ou
Rem
emb
er
3.V
isua
lizin
g th
eir
grap
hs is
oft
en a
goo
d w
ay t
o re
mem
ber
the
diff
eren
ce b
etw
een
mat
hem
atic
al e
quat
ions
.How
can
you
r kn
owle
dge
of t
he g
raph
s of
exp
onen
tial
equ
atio
nsfr
om L
esso
n 10
-1 h
elp
you
to r
emem
ber
that
equ
atio
ns o
f th
e fo
rm y
!a(
1 (
r)t
repr
esen
t ex
pone
ntia
l gro
wth
,whi
le e
quat
ions
of
the
form
y!
a(1
%r)
tre
pres
ent
expo
nent
ial d
ecay
?S
ampl
e an
swer
:If
a$
0,th
e gr
aph
of y
!ab
xis
alw
ays
incr
easi
ng if
b
$1
and
is a
lway
s de
crea
sing
if 0
'b
'1.
Sin
ce r
is a
lway
s a
posi
tive
num
ber,
if b
!1
#r,
the
base
will
be
grea
ter
than
1 a
nd t
he f
unct
ion
will
be in
crea
sing
(gr
owth
),w
hile
if b
!1
"r,
the
base
will
be
less
tha
n 1
and
the
func
tion
will
be
decr
easi
ng (
deca
y).
©G
lenc
oe/M
cGra
w-H
ill60
8G
lenc
oe A
lgeb
ra 2
Eff
ectiv
e A
nnua
l Yie
ldW
hen
inte
rest
is c
ompo
unde
d m
ore
than
onc
e pe
r ye
ar,t
he e
ffec
tive
ann
ual
yiel
d is
hig
her
than
the
ann
ual i
nter
est
rate
.The
eff
ecti
ve a
nnua
l yie
ld,E
,is
the
inte
rest
rat
e th
at w
ould
giv
e th
e sa
me
amou
nt o
f in
tere
st if
the
inte
rest
wer
e co
mpo
unde
d on
ce p
er y
ear.
If P
dolla
rs a
re in
vest
ed f
or o
ne y
ear,
the
valu
e of
the
inve
stm
ent
at t
he e
nd o
f th
e ye
ar is
A!
P(1
(E
).If
Pdo
llars
are
inve
sted
for
one
yea
r at
a n
omin
al r
ate
rco
mpo
unde
d n
tim
es p
er y
ear,
the
valu
e of
the
inve
stm
ent
at t
he e
nd o
f th
e ye
ar is
A!
P!1
(& nr & "n .S
etti
ng
the
amou
nts
equa
l and
sol
ving
for
Ew
ill p
rodu
ce a
for
mul
a fo
r th
e ef
fect
ive
annu
al y
ield
.
P(1
(E
) !P!1
(& nr & "n
1 (
E!
!1 (
& nr & "n
E!
!1 (
& nr & "n%
1
If c
ompo
undi
ng is
con
tinu
ous,
the
valu
e of
the
inve
stm
ent
at t
he e
nd o
f on
eye
ar is
A!
Per .
Aga
in s
et t
he a
mou
nts
equa
l and
sol
ve fo
r E
.A fo
rmul
a fo
rth
e ef
fect
ive
annu
al y
ield
und
er c
onti
nuou
s co
mpo
undi
ng is
obt
aine
d.
P(1
(E
) !Pe
r
1 (
E!
er
E!
er%
1
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
10-6
10-6
Fin
d t
he
effe
ctiv
ean
nu
al y
ield
of
an i
nve
stm
ent
mad
e at
7.5%
com
pou
nd
ed m
onth
ly.
r!
0.07
5n
!12
E!
!1 (
&0.10 275 &
"12%
1 *
7.7
6%
Fin
d t
he
effe
ctiv
ean
nu
al y
ield
of
an i
nve
stm
ent
mad
e at
6.25
% c
omp
oun
ded
con
tin
uou
sly.
r!
0.06
25E
!e0
.062
5%
1 *
6.4
5%
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Fin
d t
he
effe
ctiv
e an
nu
al y
ield
for
eac
h i
nve
stm
ent.
1.10
% c
ompo
unde
d qu
arte
rly
10.3
8%2.
8.5%
com
poun
ded
mon
thly
8.84
%
3.9.
25%
com
poun
ded
cont
inuo
usly
9.69
%4.
7.75
% c
ompo
unde
d co
ntin
uous
ly8.
06%
5.6.
5% c
ompo
unde
d da
ily (
assu
me
a 36
5-da
y ye
ar)
6.72
%
6.W
hich
inve
stm
ent
yiel
ds m
ore
inte
rest
—9%
com
poun
ded
cont
inuo
usly
or
9.2%
com
poun
ded
quar
terl
y?9.
2% q
uart
erly