7.6 Solve Exponential and Logarithmic Equations 515 Solve Exponential and Logarithmic Equations 7.6 Key Vocabulary • exponential equation • logarithmic equation • extraneous solution, p. 52 Exponential equations are equations in which variable expressions occur as exponents. The result below is useful for solving certain exponential equations. E XAMPLE 1 Solve by equating exponents Solve 4 x 5 1 1 } 2 2 x 2 3 . 4 x 5 1 1 } 2 2 x 2 3 Write original equation. (2 2 ) x 5 (2 21 ) x 2 3 Rewrite 4 and 1 } 2 as powers with base 2. 2 2x 5 2 2x 1 3 Power of a power property 2x 52 x 1 3 Property of equality for exponential equations x 5 1 Solve for x. c The solution is 1. CHECK Check the solution by substituting it into the original equation. 4 1 0 1 1 } 2 2 1 2 3 Substitute 1 for x. 4 0 1 1 } 2 2 22 Simplify. 4 5 4 ✓ Solution checks. ✓ GUIDED PRACTICE for Example 1 Solve the equation. 1. 9 2x 5 27 x 2 1 2. 100 7x 1 1 5 1000 3x 2 2 3. 81 3 2 x 5 1 1 } 3 2 5x 2 6 KEY CONCEPT For Your Notebook Property of Equality for Exponential Equations Algebra If b is a positive number other than 1, then b x 5 b y if and only if x 5 y. Example If 3 x 5 3 5 , then x 5 5. If x 5 5, then 3 x 5 3 5 . Before You studied exponential and logarithmic functions. Now You will solve exponential and logarithmic equations. Why? So you can solve problems about astronomy, as in Example 7. 2A.11.A, 2A.11.C, 2A.11.D, 2A.11.F TEKS
Transcript
7.6 Solve Exponential and Logarithmic Equations 515
Solve Exponential andLogarithmic Equations7.6
Key Vocabulary• exponential
equation• logarithmic
equation• extraneous
solution, p. 52
Exponential equations are equations in which variable expressions occur asexponents. The result below is useful for solving certain exponential equations.
E X A M P L E 1 Solve by equating exponents
Solve 4x 5 1 1}2 2x 2 3
.
4x 5 1 1}2 2x 2 3
Write original equation.
(22)x5 (221)x 2 3
Rewrite 4 and 1}2
as powers with base 2.
22x 5 22x 1 3 Power of a power property
2x 5 2x 1 3 Property of equality for exponential equations
x 5 1 Solve for x.
c The solution is 1.
CHECK Check the solution by substituting it into the original equation.
Algebra If b is a positive number other than 1, then bx 5 b y if and onlyif x 5 y.
Example If 3x 5 35, then x 5 5. If x 5 5, then 3x 5 35.
Before You studied exponential and logarithmic functions.
Now You will solve exponential and logarithmic equations.
Why? So you can solve problems about astronomy, as in Example 7.
2A.11.A, 2A.11.C,2A.11.D, 2A.11.F
TEKS
516 Chapter 7 Exponential and Logarithmic Functions
E X A M P L E 3 Use an exponential model
CARS You are driving on a hot day whenyour car overheats and stops running. Itoverheats at 2808F and can be driven againat 2308F. If r 5 0.0048 and it is 808F outside,how long (in minutes) do you have to waituntil you can continue driving?
Solution
T 5 (T0 2 TR)e2rt 1 TR Newton’s law of cooling
230 5 (280 2 80)e20.0048t 1 80 Substitute for T, T0, TR, and r.
150 5 200e20.0048t Subtract 80 from each side.
0.75 5 e20.0048t Divide each side by 200.
ln 0.75 5 ln e20.0048t Take natural log of each side.
20.2877 ø 20.0048t ln ex 5 loge ex 5 x
60 ø t Divide each side by 20.0048.
c You have to wait about 60 minutes until you can continue driving.
When it is not convenient to write each side of an exponential equation using thesame base, you can solve the equation by taking a logarithm of each side.
E X A M P L E 2 Take a logarithm of each side
Solve 4x 5 11.
4x 5 11 Write original equation.
log4 4x 5 log4 11 Take log4 of each side.
x 5 log4 11 logb bx 5 x
x 5log 11}log 4
Change-of-base formula
x ø 1.73 Use a calculator.
c The solution is about 1.73. Check this in the original equation.
ANOTHER WAY
For an alternativemethod for solving theproblem in Example 2,turn to page 523 forthe Problem SolvingWorkshop.
NEWTON’S LAW OF COOLING An important application of exponential equationsis Newton’s law of cooling. This law states that for a cooling substance with initialtemperature T0, the temperature T after t minutes can be modeled by
T 5 (T0 2 TR)e2rt 1 TR
where TR is the surrounding temperature and r is the substance’s cooling rate.
✓ GUIDED PRACTICE for Examples 2 and 3
Solve the equation.
4. 2x 5 5 5. 79x 5 15 6. 4e20.3x 2 7 5 13
7.6 Solve Exponential and Logarithmic Equations 517
SOLVING LOGARITHMIC EQUATIONS Logarithmic equations are equations thatinvolve logarithms of variable expressions. You can use the following property tosolve some types of logarithmic equations.
log5 (4 p 4 2 7) 0 log5 (4 1 5) Substitute 4 for x.
log5 9 5 log5 9 ✓ Solution checks.
EXPONENTIATING TO SOLVE EQUATIONS The property of equality for exponentialequations on page 515 implies that if you are given an equation x 5 y, thenyou can exponentiate each side to obtain an equation of the form bx 5 by. Thistechnique is useful for solving some logarithmic equations.
E X A M P L E 5 Exponentiate each side of an equation
Solve log4 (5x 2 1) 5 3.
log4 (5x 2 1) 5 3 Write original equation.
4log4(5x 2 1) 5 43 Exponentiate each side using base 4.
5x 2 1 5 64 blogb x 5 x
5x 5 65 Add 1 to each side.
x 5 13 Divide each side by 5.
c The solution is 13.
CHECK log4 (5x 2 1) 5 log4 (5 p 13 2 1) 5 log4 64
Because 43 5 64, log4 64 5 3. ✓
KEY CONCEPT For Your Notebook
Property of Equality for Logarithmic Equations
Algebra If b, x, and y are positive numbers with b Þ 1, then logb x 5 logb yif and only if x 5 y.
Example If log2 x 5 log2 7, then x 5 7. If x 5 7, then log2 x 5 log2 7.
518 Chapter 7 Exponential and Logarithmic Functions
EXTRANEOUS SOLUTIONS Because the domain of a logarithmic functiongenerally does not include all real numbers, be sure to check for extraneoussolutions of logarithmic equations. You can do this algebraically or graphically.
✓ GUIDED PRACTICE for Examples 4, 5, and 6
Solve the equation. Check for extraneous solutions.
log 1000 0 3 Because log (240) and log (225) arenot defined, 25 is not a solution.
3 5 3 ✓
So, 25 is a solution.
Graph Graph y 5 log 8x 1 log (x 2 20) and y 5 3in the same coordinate plane. The graphsintersect only once, when x 5 25. So, 25 isthe only solution.
c The correct answer is C. A B C D
What is (are) the solution(s) of log 8x 1 log (x 2 20) 5 3?
A 25, 25 B 5 C 25 D 5, 25
IntersectionX=25 Y=3
ELIMINATE CHOICES
Instead of solving theequation in Example 6directly, you cansubstitute each possibleanswer into theequation to see whetherit is a solution.
7.6 Solve Exponential and Logarithmic Equations 519
E X A M P L E 7 Use a logarithmic model
ASTRONOMY The apparent magnitude of a staris a measure of the brightness of the star as itappears to observers on Earth. The apparentmagnitude M of the dimmest star that can beseen with a telescope is given by the function
M 5 5 log D 1 2
where D is the diameter (in millimeters) of thetelescope’s objective lens. If a telescope can reveal starswith a magnitude of 12, what is the diameter of its objective lens?
Solution
M 5 5 log D 1 2 Write original equation.
12 5 5 log D 1 2 Substitute 12 for M.
10 5 5 log D Subtract 2 from each side.
2 5 log D Divide each side by 5.
102 5 10log D Exponentiate each side using base 10.
100 5 D Simplify.
c The diameter is 100 millimeters.
at classzone.com
✓ GUIDED PRACTICE for Example 7
11. WHAT IF? Use the information from Example 7 to find the diameter of theobjective lens of a telescope that can reveal stars with a magnitude of 7.
1. VOCABULARY Copy and complete: The equation 5x 5 8 is an example ofa(n) ? equation.
2. ★ WRITING When do logarithmic equations have extraneous solutions?
44. ★ MULTIPLE CHOICE What is the solution of 3 log8 (2x 1 7) 1 8 5 10?
A 21.5 B 21.179 C 4 D 4.642
ERROR ANALYSIS Describe and correct the error in solving the equation.
45. 3x 1 1 5 6x
log3 3x 1 1 5 log3 6x
x 1 1 5 x log3 6
x 1 1 5 2x
1 5 x
46. log3 10x 5 5
elog3 10x 5 e5
10x 5 e5
x 5 e5}10
47. ★ OPEN-ENDED MATH Give an example of an exponential equation whose only solution is 4 and an example of a logarithmic equation whose only solution is 23.
CHALLENGE Solve the equation.
48. 3x 1 4 5 62x 2 5 49. 103x 2 8 5 25 2 x
50. log2 (x 1 1) 5 log8 3x 51. log3 x 5 log9 6x
52. 22x 2 12 p 2x 1 32 5 0 53. 52x 1 20 p 5x 2 125 5 0
TAKS REASONING
TAKS REASONING
5 TAKS PRACTICE AND REASONING
7.6 Solve Exponential and Logarithmic Equations 521
GREECEUSA JAPAN
Ocotillo Wells, CAMay 20, 2005R = 4.1
AthensSept. 7, 1999R = 5.9
FukuokaMarch 20, 2005R = 6.6
54. COOKING You are cooking beef stew. When you take the beef stew off thestove, it has a temperature of 2008F. The room temperature is 758F and thecooling rate of the beef stew is r 5 0.054. How long (in minutes) will it take tocool the beef stew to a serving temperature of 1008F?
55. THERMOMETER As you are hanging an outdoor thermometer, its readingdrops from the indoor temperature of 758F to 378F in one minute. If thecooling rate is r 5 1.37, what is the outdoor temperature?
56. COMPOUND INTEREST You deposit $100 in an account that pays 6% annualinterest. How long will it take for the balance to reach $1000 for each givenfrequency of compounding?
a. Annual b. Quarterly c. Daily
57. RADIOACTIVE DECAY One hundred grams of radium are stored in acontainer. The amount R (in grams) of radium present after t years can bemodeled by R 5 100e20.00043t. After how many years will only 5 grams ofradium be present?
58. ★ MULTIPLE CHOICE You deposit $800 in an account that pays 2.25% annualinterest compounded continuously. About how long will it take for thebalance to triple?
A 24 years B 36 years
C 48.8 years D 52.6 years
59. MULTIPLE REPRESENTATIONS The Richter scale is used for measuringthe magnitude of an earthquake. The Richter magnitude R is given by thefunction
R 5 0.67 log (0.37E) 1 1.46
where E is the energy (in kilowatt-hours) released by the earthquake.
a. Making a Graph Graph the function using a graphing calculator. Useyour graph to approximate the amount of energy released by eachearthquake indicated in the diagram above.
b. Solving Equations Write and solve a logarithmic equation to find theamount of energy released by each earthquake in the diagram.
PROBLEM SOLVING
EXAMPLE 3
on p. 516for Exs. 54–58
EXAMPLE 7
on p. 519for Ex. 59
TAKS REASONING
522 Chapter 7 Exponential and Logarithmic FunctionsEXTRA PRACTICE for Lesson 7.6, p. 1016 ONLINE QUIZ at classzone.com
60. ★ EXTENDED RESPONSE If X-rays of a fixed wavelength strike a materialx centimeters thick, then the intensity I(x) of the X-rays transmitted throughthe material is given by I(x) 5 I0e2μx, where I0 is the initial intensity and μ is anumber that depends on the type of material and the wavelength of theX-rays. The table shows the values of μ for various materials. These μ-valuesapply to X-rays of medium wavelength.
Material Aluminum Copper Lead
Value of μ 0.43 3.2 43
a. Find the thickness of aluminum shielding that reduces the intensityof X-rays to 30% of their initial intensity. (Hint: Find the value of x forwhich I(x) 5 0.3I0.)
b. Repeat part (a) for copper shielding.
c. Repeat part (a) for lead shielding.
d. Reasoning Your dentist puts a lead apron on you before taking X-raysof your teeth to protect you from harmful radiation. Based on yourresults from parts (a)–(c), explain why lead is a better material to usethan aluminum or copper.
61. CHALLENGE You plant a sunflowerseedling in your garden. Theseedling’s height h (in centimeters)after t weeks can be modeled bythe function below, which is calleda logistic function.
h(t) 5 256}1 1 13e20.65t
Find the time it takes thesunflower seedling to reach aheight of 200 centimeters. Weeks
Heig
ht (c
m)
2 4 6 8
200
100
00 t
h
TAKS REASONING
62. TAKS PRACTICE Which list shows the functions in order from the widestgraph to the narrowest graph? TAKS Obj. 5
A y 5 25x2, y 5 22}3
x2, y 5 5}6
x2, y 5 8x2
B y 5 22}3
x2, y 5 5}6
x2, y 5 25x2, y 5 8x2
C y 5 5}6
x2, y 5 22}3
x2, y 5 8x2, y 5 25x2
D y 5 8x2, y 5 5}6
x2, y 5 22}3
x2, y 5 25x2
63. TAKS PRACTICE In the diagram, m∠ 2 5 m∠ 3.What is m∠ 1? TAKS Obj. 6
F 1368 G 1648
H 1748 J 1948
MIXED REVIEW FOR TAKS
REVIEW
Lesson 4.1;TAKS Workbook
REVIEW
Skills ReviewHandbook p. 994;TAKS Workbook
TAKS PRACTICE at classzone.com
658728
9583
2
1
Using Alternative Methods 523
Using a Graph You can also use a graph to solve the equation.
STEP 1 Enter the functions y 5 4x and y 5 11 into agraphing calculator.
STEP 2 Graph the functions. Use the intersect featureto find the intersection point of the graphs.The graphs intersect at about (1.73, 11).
c The solution of 4x 5 11 is about 1.73.
M E T H O D 2
IntersectionX=1.7297158 Y=11
Y1=4^XY2=11Y3=Y4=Y5=Y6=Y7=
Another Way to Solve Examples 2 and 7, pp. 516 and 519
PRO B L E M 1 Solve the following exponential equation: 4x 5 11.
Using a Table One way to solve the equation is to make a table of values.
STEP 1 Enter the function y 5 4x intoa graphing calculator.
Y1=4^XY2=Y3=Y4=Y5=Y6=Y7=
STEP 2 Create a table of values for thefunction.
Y189.189610.55612.12613.929
X1.51.61.71.81.9X=1.7
STEP 3 Scroll through the table to find when y 5 11. The table in Step 2 showsthat y 5 11 between x 5 1.7 and x 5 1.8.
c The solution of 4x 5 11 is between 1.7 and 1.8.
M E T H O D 1
MULTIPLE REPRESENTATIONS In Examples 2 and 7 on pages 516 and 519,respectively, you solved exponential and logarithmic equations algebraically.You can also solve such equations using tables and graphs.
LESSON 7.6
ALTERNATIVE METHODSALTERNATIVE METHODSUsingUsing
Use a viewingwindow of 0 ≤ x ≤ 5and 0 ≤ y ≤ 20.
a.5, a.6, 2A.11.D,2A.11.F
TEKS
524 Chapter 7 Exponential and Logarithmic Functions524 Chapter 7 Exponential and Logarithmic Functions
ASTRONOMY The apparentmagnitude of a star is a measureof the brightness of the star as itappears to observers on Earth.The apparent magnitude M of thedimmest star that can be seen witha telescope is given by the function
M 5 5 log D 1 2
where D is the diameter (in millimeters) of the telescope’s objective lens.If a telescope can reveal stars with a magnitude of 12, what is the diameterof its objective lens?
Using a Table Notice that the problem requires solving the following logarithmicequation:
5 log D 1 2 5 12
One way to solve this equation is to make a table of values. You can use agraphing calculator to make the table.
STEP 1 Enter the function y 5 5 log x 1 2 into agraphing calculator.
STEP 2 Create a table of values for the function.Make sure that the x-values are in thedomain of the function (x > 0).
STEP 3 Scroll through the table of values to findwhen y 5 12.
c To reveal stars with a magnitude of 12, a telescope must have an objective lenswith a diameter of 100 millimeters.
M E T H O D 1
PRO B L E M 2
Y111.95611.9781212.02212.043
X9899100101102X=100
Y123.50514.38565.01035.4949
X12345X=1
Y1=5*log(X)+2Y2=Y3=Y4=Y5=Y6=Y7=
The table shows that y 5 12when x 5 100.
Using Alternative Methods 525
Using a Graph You can also use a graph to solve the equation 5 log D 1 2 5 12.
STEP 1 Enter the functions y 5 5 log x 1 2 andy 5 12 into a graphing calculator.
STEP 2 Graph the functions. Use the intersectfeature to find the intersection point of thegraphs. The graphs intersect at (100, 12).
c To reveal stars with a magnitude of 12, a telescope must have an objective lenswith a diameter of 100 millimeters.
PR AC T I C E
EXPONENTIAL EQUATIONS Solve the equationusing a table and using a graph.
1. 8 2 2e3x 5 214
2. 7 2 105 2 x 5 29
3. e5x 2 8 1 3 5 15
4. 1.6(3)24x 1 5.6 5 6
LOGARITHMIC EQUATIONS Solve the equationusing a table and using a graph.
5. log2 5x 5 2
6. log (23x 1 7) 5 1
7. 4 ln x 1 6 5 12
8. 11 log (x 1 9) 2 5 5 8
9. ECONOMICS From 1998 to 2003, the UnitedStates gross national product y (in billions ofdollars) can be modeled by y 5 8882(1.04)x
where x is the number of years since 1998.Use a table and a graph to find the year whenthe gross national product was $10 trillion.
10. WRITING In Method 1 of Problem 1 onpage 523, explain how you could use a table tofind the solution of 4x 5 11 more precisely.
11. WHAT IF? In Problem 2 on page 524, supposethe telescope can reveal stars of magnitude 14.Find the diameter of the telescope’s objectivelens using a table and using a graph.
12. FINANCE You deposit $5000 in an accountthat pays 3% annual interest compoundedquarterly. How long will it take for the balanceto reach $6000? Solve the problem using a tableand using a graph.
13. OCEANOGRAPHY The density d (in grams percubic centimeter) of seawater with a salinityof 30 parts per thousand is related to the watertemperature T (in degrees Celsius) by thefollowing equation:
d 5 1.0245 2 e0.1226T 2 7.828
For deep water in the South Atlantic Ocean offAntarctica, d 5 1.0241 g/cm3. Use a table and agraph to find the water’s temperature.
IntersectionX=100 Y=12
Y1=5*log(X)+2Y2=12Y3=Y4=Y5=Y6=Y7=
M E T H O D 2
Use a viewing window of0 ≤ x ≤ 150 and 0 ≤ y ≤ 20.
526 Chapter 7 Exponential and Logarithmic Functions526 Chapter 7 Exponential and Logarithmic Functions
Solve Exponential andLogarithmic Inequalities
CARS Your family purchases a new car for $20,000. Its value decreases by 15%each year. During what interval of time does the car’s value exceed $10,000?
Solution
Let y represent the value of the car (in dollars) x years after it is purchased. Afunction relating x and y is y 5 20,000(1 2 0.15)x, or y 5 20,000(0.85)x. To findthe values of x for which y > 10,000, solve the inequality 20,000(0.85)x > 10,000.
METHOD 1 Use a table
STEP 1 Enter the function y 5 20,000(0.85)x intoa graphing calculator. Set the startingx-value of the table to 0 and the stepvalue to 0.1.
STEP 2 Use the table feature to create a table ofvalues. Scrolling through the table showsthat y > 10,000 when 0 ≤ x ≤ 4.2.
c The car value exceeds $10,000 for about the first4.2 years after it is purchased.
To check the solution’s reasonableness, note thaty ø 10,440 when x 5 4 and y ø 8874 when x 5 5.So, 4 < x < 5, which agrees with the solutionobtained above.
METHOD 2 Use a graph
Graph y 5 20,000(0.85)x and y 5 10,000 in thesame viewing window. Set the viewing windowto show 0 ≤ x ≤ 8 and 0 ≤ y ≤ 25,000. Using theintersect feature, you can determine that thegraphs intersect when x ø 4.27.
c The car value exceeds $10,000 for about the first 4.27 years after it is purchased.
E X A M P L E 1 Solve an exponential inequality
GOAL Solve exponential and logarithmic inequalities using tables and graphs.
In the Problem Solving Workshop on pages 523–525, you learned how to solveexponential and logarithmic equations using tables and graphs. You can use thesesame methods to solve exponential and logarithmic inequalities.
Use after Lesson 7.6
IntersectionX=4.2650243 Y=10000
Y11044010272101069943.39783
X44.14.24.34.4X=4.2
The graph of y 5 20,000(0.85)x is above thegraph of y 5 10,000 when 0 ≤ x < 4.27.
Extension
TABLE SETUP TblStart=0
nTbl=0.1Indpnt: Auto AskDepend: Auto Ask
2A.11.E,2A.11.F
TEKS
Extension: Solve Exponential and Logarithmic Inequalities 527
E X A M P L E 2 Solve a logarithmic inequality
Solve log2 x ≤ 2.
Solution
METHOD 1 Use a table
STEP 1 Enter the function y 5 log2 x into a
graphing calculator as y 5log x}log 2
.
STEP 2 Use the table feature to create a table ofvalues. Identify the x-values for whichy ≤ 2. These x-values are given by 0 < x ≤ 4.
c The solution is 0 < x ≤ 4.
METHOD 2 Use a graph
Graph y 5 log2 x and y 5 2 in the same viewingwindow. Using the intersect feature, you candetermine that the graphs intersect when x 5 4.
c The solution is 0 < x ≤ 4.
Y1011.58522.3219
X12345X=4
IntersectionX=4 Y=2
Make sure that the x-values are reasonableand in the domain of the function (x > 0).
The graph of y 5 log2 x is on or belowthe graph of y 5 2 when 0 < x ≤ 4.
Solve the exponential inequality using a table and using a graph.
Solve the logarithmic inequality using a table and using a graph.
7. log3 x ≥ 3 8. log5 x < 2 9. log6 x 1 9 ≤ 11
10. 2 log4 x 2 1 > 4 11. 24 log2 x > 220 12. 0 ≤ log7 x ≤ 1
13. FINANCE You deposit $1000 in an account that pays 3.5% annual interestcompounded monthly. When is your balance at least $1200?
14. RATES OF RETURN An investment that earns a rate of return r doubles in
value in t years, where t 5 ln 2}ln (1 1 r)
and r is expressed as a decimal. What
rates of return will double the value of an investment in less than 10 years?
PRACTICE
EXAMPLE 1
on p. 526for Exs. 1–6
EXAMPLE 2
on p. 527for Exs. 7–12
Y1=log(X)/log(2)Y2=Y3=Y4=Y5=Y6=
528 Chapter 7 Exponential and Logarithmic Functions
7.7 Model Data with an Exponential FunctionMATERIALS • 100 pennies • cup • graphing calculator
Q U E S T I O N How can you model data with an exponential function?
E X P L O R E Collect and record data
STEP 1 Make a table
Make a table like the one shown to record your results.
Number of toss, x 0 1 2 3 4 5 6 7
Number of pennies remaining, y ? ? ? ? ? ? ? ?
STEP 2 Perform an experiment
Record the initial number of pennies Remove all of the pennies showingin the table, and place the pennies in “heads.” Count the number of penniesa cup. Shake the pennies, and then remaining, and record this number inspill them onto a flat surface. the table.
STEP 3 Continue collecting data
Repeat Step 2 with the remaining pennies until there are no pennies left toreturn to the cup.
D R A W C O N C L U S I O N S Use your observations to complete these exercises
1. What is the initial number of pennies? By what percent would you expectthe number of pennies remaining to decrease after each toss?
2. Use your answers from Exercise 1 to write an exponential function thatshould model the data in the table.
3. Use a graphing calculator to make a scatter plot of the data pairs (x, y).In the same viewing window, graph your function from Exercise 2. Is thefunction a good model for the data? Explain.
4. Use the calculator’s exponential regression feature to find an exponentialfunction that models the data. Compare this function with the functionyou wrote in Exercise 2.
Use before Lesson 7.7ACTIVITYACTIVITYInvestigating Algebra
