1

Honors Math 3 Unit 4 Exponential and Logs NAME ___________________________

Day Date Topic(s) /Classwork Homework

1

Solving Radical Equations p. 2, odds p. 17 p. 3, evens

2 Rules for Domain Inverses of Functions

p. 8: #’s 1-8

3

Graphing Exponentials with transformations p. 11-12: #’s 1-8

4 Solving exponential equations with like bases More Practice with Laws of Exponents (Advanced)

p. 13-14: Do evens

5 Quiz 1 (Days 1-4) Introduction to Logs, Graphing logs

p. 16: #’s 1-6

6

Exponential Form to Log Form (vice versa) Evaluating logs Solving Log Equations (single logs)

p. 18: #’s 1-22

7 Group Quiz Test Review: p. 20-22

Finish p. 20-22 p. 34: Do problems with *

8 Unit 5 Test (Part 1) Delta Math HW Online

9 Properties of Logs Solving Log equations

p. 26 #’s 1-40

10 Natural Logs, Common Logs Solving with base e Odds on p. 29-30

Evens p. 29-30

11 Quiz 2 Change of Base Solving with Unlike bases

p. 31: #’s 1-25

12 Applications and review p. 33: #’s 1-8

13 Group Quiz Review: p. 34-37

p. 34-37

14 Unit 5 TEST (Part 2) Delta Math Online Problem Set

2

Day 1 CW: Solving Radical Equations: EXTRANEOUS SOLUTIONS: answers that do not work when put back into original problem. *only check for extraneous solutions when variable is under radical and index is even *If variable is under radical, isolate radical and raise both sides to the index value. *If variable in NOT under radical, isolate variable, simplify.

1. 𝑥 + 1 = 𝑥√5 2. √2𝑥 + 3 = 𝑥

3. √4𝑥 + 28 − 3√2𝑥 = 0 4. 2𝑥 + 7 = −𝑥√2

5. √7𝑥 − 3 − 2𝑥 + 3 = 0 6. √𝑥 + 8 − √𝑥 + 35 = −3

7. √2𝑦 − 3 − √2𝑦 + 3 = −1 8. √𝑦 − 53 − 2 = −4

9. √𝑥 + 11 − √15 + 2𝑥 = 1 10. −2√2𝑥 − 15

+ 4 = 0

3

Day 1 HW: Solving Equations Containing Radicals Solve each equation. (#4 – 16 Check answers for extraneous solutions! Except #5,10)

1. 0537 x 2. 634 xx

3. 2318 xx 4. 058 x

5. 473 y 6. 0234 x

7. 2158 n 8. 6841 t

9. 712274 v 10. 32563 u

11. 10246 xx 12. 20749 uu

13. 39 kk 14. 8610 xx

15. 972 xx 16. 052234 2 xxx

4

Day 2 Notes: RULES FOR DOMAIN: Write answers using interval notation. (1) If x is in the numerator and raised to a positive integral exponent the domain: (−∞, ∞) all reals a. f(x) = x2 D: _______________________________________

b. f(x) = 8

4

3

x

D: _______________________________________ (2) If x is in the denominator, x cannot be any value that will make the denominator zero.

a. f(x) = x

1

D: _________________________________________

b. f(x) = 3x

x

D: __________________________________________

c. f(x) = )4)(2(

1

xx

x

D: __________________________________________

(3) If x is inside an even root, values of x are restricted to ones that will make the radicand 0.

a. f(x) = x D: ____________________________________________

b. f(x) = 3x D: ____________________________________________

c. f(x) = √2𝑥 − 54

D: ____________________________________________

d. 𝑦 = √2 − 4𝑥 D: ____________________________________________ (4) If x is in an even root and in the denominator, values of x are restricted to the one that will make the radicand > 0.

a. f(x) = x

1

D: _____________________________________________

b. f(x) = 5x

x

D: ______________________________________________

c. 𝑦 =𝑥+3

√7−𝑥4 D: _____________________________________________

Find the Domain.

1. 𝑦 = √3𝑥3

2. 𝑦 = 1

𝑥2 + 1 3. 𝑦 =𝑥−2

𝑥2−2𝑥−15

5

Day 2 CW: Inverse Functions

Examples:

1) Given the t-chart for 𝑓(𝑥), find the t-chart for 𝑓−1(𝑥). 2) Given graph of 𝑓(𝑥), graph 𝑓−1(𝑥).

Find the inverse.

3) 2)( xxf 4) 𝑓(𝑥) = 𝑥2 + 3 5) 𝑦 = 3𝑥2 − 2

Graph the following function, find the domain and range, then graph its inverse and find the domain and range.

6) 𝑦 = √𝑥

D: ________________

R: ________________

𝑓−1(𝑥) = _____________

D: ________________

R: ________________

)(xf )(1 xf

x y x y

-5 0

-3 -2

1 5

2 -1

3 3

5 6

6

7) 𝑦 = √𝑥 + 3

D: ________________

R: ________________

𝑓−1(𝑥) = _____________

D: _______________

R: ________________

8) 𝑦 = √𝑥 + 3

D: ________________

R: ________________

f-1(x) = _____________

D: _______________

R: ________________

Find the inverse of 𝑓(𝑥).

9) 𝑓(𝑥) = 𝑥3 + 2 10) 𝑓(𝑥) = 𝑥2

3 11) 𝑓(𝑥) = (𝑥 − 4)2

7

Find the inverse of each function and graph both the function and its inverse on the same grid. Then state

the domain of both the function and its inverse.

12. 𝒇(𝒙) = √𝒙 − 𝟑 D:_____________ R:______________

𝒇−𝟏(𝒙) = _______________ D:_____________ R:______________

13. 𝒇(𝒙) = −𝒙𝟐 − 𝟐 D:_____________ R:______________

𝒇−𝟏(𝒙) = _______________ D:_____________ R:______________

8

Day 2 HW. Find the inverse of each function.

1. 𝑦 = 𝑥2 − 9 2. 𝑦 = (3 − 𝑥)2 − 1 3. 𝑦 = √𝑥 − 1 + 3

4. 𝑓(𝑥) = 3𝑥2 − 5 5. 𝑓(𝑥) = 2𝑥2

5+ 1 6. 𝑓(𝑥) = √𝑥 + 2 − 4

Find the inverse of the given function and graph both the function and its inverse on the same grid. Then

state the domain of both the function and its inverse.

7. 𝒇(𝒙) = √𝒙 − 𝟒

D:_____________ R:______________

𝒇−𝟏(𝒙) = _______________

D:_____________ R:______________

8. 𝒇(𝒙) = 𝒙𝟐 + 𝟑

D:_____________ R:______________

𝒇−𝟏(𝒙) = _______________

D:_____________ R:______________

9

Day 3 CW: Graphing Exponential Functions

Complete the tables, then graph.

Ex1. y = 2x

Ex 2. y = 4x

Ex 3. y = (1/2)x

𝑦 = 𝑎(2)𝑏(𝑥−ℎ) + 𝑘

If a is negative: Reflect over the __________ If b is negative: Reflect over the _____________

h: __________________ or _______________ k: ___________ or _____________

|a| > 1 Vertical ______________ 0 < |a| < 1 Vertical _______________

x y= 2x Y

-4

-3

-2 y = 2-2 ¼

-1 y = 2-1 ½

0 y = 20 1

1 y = 21 2

2 y = 22 4

3

4

5

Domain: Range: Y-intercept: X-intercept: Asymptote:

x y= (1/2)x y

-2 -1

0 y = (1/2)0 1

2

Domain: Range: Y-intercept: X-intercept: Asymptote:

x y= (4)x y -3

-2

-1 0 y = (4)0

1 2

3

Domain: Range: Y-intercept: X-intercept: Asymptote:

10

Day 3 Examples: Graph, describe transformations, state the domain & range

1. 12 xy Transformation: 2. 12 xy Transformation:

3. 32 2 xy Transformations: 4. xy 2 Transformation:

5. xy

2 Transformations: 6. x

y 24 Transformation:

Domain: Range:

Domain: Range:

Domain: Range:

Domain: Range:

Domain: Range:

Domain: Range:

11

7. 221

xy Transformations: 8. x

y

23 Transformations:

9. xy 5 10. 25 1 xy Transformations:

Day 3 HW: Graphing Exponential Functions Worksheet

1. 13 xy 2. 23 xy

Domain: Range:

Domain: Range:

Domain: Range:

Domain: Range:

Domain: Range:

Domain: Range:

12

3. xy

3 4.

y 2x2 3

5. 42 x

y 6. xy

32

7. 423

xy 8. 132

2

xy

Domain: Range:

Domain: Range:

Domain: Range:

Domain: Range:

Domain: Range:

Domain: Range:

13

Day 4 CW/HW: I. Solve. 1. 53 = 52x – 1 2. 9x = 3x + 4 3. 5x – 1 = 1252x + 3 4. 23x – 1 = 4x + 2

5. 82x – 2 = 42 – x 6. 1

3

33

1

x

x

7. 3

42

49

17

x

x 8. 63

3

25

1125

x

x

9. 33

8

1

32

1

xx

10. 4

12

2

18

x

x 11. 4x = 2x + 3 12. 53x – 1 = 25 x + 4

13. 3x – 1 = 27 2x +3 14. 1252x – 2 = 253 – x 15. 1

3

66

1

x

x

16. 3

42

100

110

x

x

17. 63

3

4

18

x

x 18. 42

8

1

32

1

xx

19. 5

13

2

14

x

x 20. 252x = 5x + 6

21. 6x + 1 = 36x – 1 22. 10x – 1 = 1004 – x 23. 27

13 x 24. 1255 x

25. 82 + x = 2 26. 4 1 - x = 8 27. 272x – 1 = 3 28. 7749 2 x

14

II. Simplify & express with positive exponents.

1. 2

3 2. 04 3. 822

4. 2225 5. 32

323

6. 21215

7. 3232 33 8. 1515 55 9. 1515 22

10. 22121 33 11. 2

82ba 12. 333 yx

13. (𝑥√3 − 𝑦√2)2 14. (𝑥−√8𝑦√50)√2

15. −32𝑥

45𝑦

16

4𝑥15𝑦

56

16. (𝑥3√2𝑦−2√3)

−3

𝑥4√2𝑦−4√3

15

Day 5 CW: Logarithmic Function: inverse of an exponential function Find the inverse of 𝑦 = 𝑏𝑥 Example 1: Graph on the same axis. Make a table for each.

a) y = 2x

b) xy2

log

1. )(log 22

xy 2. 22 xy log

Domain: Range: Asymptote:

Domain: Range: Asymptote:

16

Day 5 HW: Graphing Log Functions Worksheet

1. xy2

log 2.

y log2(x 3)

3.

y log2 x 3 4.

y log2(x 1) 2

5.

y log4 x 1 6.

y log4 (x 2) 3

Domain: Range: Asymptote:

Domain: Range: Asymptote:

Domain: Range: Asymptote:

Domain: Range: Asymptote:

Domain: Range: Asymptote:

Domain: Range: Asymptote:

17

xyb log ybx

Day 6 CW: Find the inverse of 𝑦 = 2𝑥.

y is called the logarithm, base 2 of x. It is written in “log form” as: A LOGARITHM is AN _______________________. Exponential Form: 𝐵𝑎𝑠𝑒𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 = 𝐴𝑛𝑠𝑤𝑒𝑟 Logarithmic Form: 𝑙𝑜𝑔𝑏𝑎𝑠𝑒(𝑎𝑛𝑠𝑤𝑒𝑟) = 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡

* Base will ALWAYS be ______________ (never = to zero, never = 1, never negative)

** Answer will ALWAYS be _______________ (never = to zero, never negative)

*** Exponent can be _________ real number

Find the exact value without using a calculator. (What power of ___ gives you ____?)

1) log5 25 2) log2 32 3) log3 27 4) log5 251

5) log5 1 6) log5 5m 7) log2 64 8) log8 64

9) log23 2 10) log2 8

1 11) log2 2 12) 1log2

A few rules about the values of b and x:

13) )2(log2 14) 2log 2 15) 2log0 16) 0log2

17) 4

2 2log 18) 𝑙𝑜𝑔5(5−3) 19) lit77log 20) 3

99

mlog 21) 7

22log

Solving logarithmic equations:

22) 213log5 x 23) 38log x 24) x9log27

25) 1316log2 x 26) 481log x

Note: log b x

Exponential Form: 32 = 9 Logarithmic Form: log 3 9 = 2

18

Day 6 CW Exercises I. Write in exponential form.

1. 38log 2 2. 481log3 3. 13

1log 3 4. 2

25

1log 5

5. 2

12log 4 6.

4

38log16 7. 01log8 8.

3

24log8

9. 481log

3

1 10. 2100log10

II. Write in logarithmic form.

1. 2525 2. 4381 3. 160 4. 283

1

5. 9

13 2 6.

64

116 3

2

7. 12525 2

3

8. 93

12

9. 32

12 5

III. Find the logarithm. 1. 1.0log10 2. 01.log10 3. 3log 27 4. 9log 27

5. 16log8 6. 2

1log 4 7.

4

1log

2

1 8. 9log3

1

9. 25log5

1 10. 10

2 2log

IV. Find x. 1. 481log x 2. 110log x 3. 216log x 4. 3125log x

5. 4log 4 x 6. 2log6 x 7. 0log3 x 8. 3

2log8 x

9. x100log10 10. x27log9 11. x8log16 12. x5.log 2

13. 4log2

x 14. 6log3

x 15. 2

13log x 16.

6

12log x

19

Day 6 HW: Write each equation in logarithmic form.

1. 53 = 125 2. 8127 3

4

Write each equation in exponential form.

3. log100.00001 = -5 4. 2

1

3

6log

2

3

Evaluate each expression. 5. 81log3 6. 0001.0log10

7. 16

1log 2 8. 27log

3

1

9. 1log 9 10. 4log8

Solve each equation.

11. 2

3log 4 x 12. 416log y

13. 38

1log a 14.

2

1log 7 n

15. 3

4log

5y 16.

6

19log 3 x

17. 47log73log 88 xx 18. 6log208log 77 xx

19. 164log19log 33 xx 20. 133log9log 1212 xx

21. 6log30log 5

2

5 x 22. xx 5log6log 4

2

4

20

Day 7 Unit 3 Part 1 Test Review:

Evaluate the following:

1. 625log5 2. 64log2 3. 1log4 4.

000,100log10 5. 2.0log5

Simplify. (No negative or zero exponents)

6. 𝑦3

2𝑥2 ∙8𝑥7

𝑦9 7. (2𝑦3

𝑥2 )5

8. 6𝑥−5𝑦−3

8𝑦−7

Find the inverse of the following:

9. 𝑓(𝑥) = 9𝑥2 − 13 10. 𝑦 = √𝑥 − 5 11. 𝑓(𝑥) =𝑥2−3

7

Express in logarithmic form.

12. 25𝑘 = 100 13. (1

3)

−3

= 27

State the transformations of each of the following from the parent graph 𝑓(𝑥) = log2 𝑥.

14. 𝑔(𝑥) = log2(𝑥 + 4) − 9 15. 𝑔(𝑥) = − log2(𝑥 + 5)

16. 𝑔(𝑥) = log2(𝑥 − 3) + 6 17. 𝑔(𝑥) = log2(−𝑥) + 6

Solve for x.

18. 43x + 1 = 8x – 1 19.

63

3

4

18

x

x

21

Solve each equation. Show all work and circle final answers. No decimal answers.

20. log125 𝑥 =2

3 21. log𝑥 3 =

1

4

Find the inverse of the given function and graph both the function and its inverse on the same grid. Then

state the domain of both the function and its inverse.

22. 𝑓(𝑥) = √𝑥 − 4

D:_____________ R:______________

𝑓−1(𝑥) = _______________

D:_____________ R:______________

23. 𝑓(𝑥) = 𝑥2 − 5

D:_____________ R:______________

𝑓−1(𝑥) = _______________

D:_____________ R:______________

22

Graph and analyze the following:

24. 𝑦 = 2𝑥+4 − 1

Transformation: ______________________________

Domain : ______________ Range: _____________

25. 𝑓(𝑥) = log2(𝑥 − 3) + 5

Transformation: _______________________________

Domain : _____________ Range: ________________

23

Day 9 CW: Properties of Logarithms:

Condensed Form Expanded Form

Product Property mnb

log

nm

bbloglog

Quotient Property n

mb

log

nmbb

loglog

Power Property

logb mp

mp

blog

Equality Property: nmthennmIf bb loglog

Examples: Use the properties of log to expand each log:

1.

log3xy 2.

log5x

y

3.

log7a2b 4.

log9a2

cb3

Use the properties of log to write each as a single log:

________________________5.

log2 x log2 y log2 z

________________________6.

3log2 x log2 y 4log2 z

________________________7.

2log xy log5 3(logz log2)

Proof of the Product Proof of the Quotient Property: Property:

nmmn

yxmn

mnb

mnbb

nbandmbthen

nyandmxLet

bbb

b

yx

yx

yx

bb

logloglog

log

loglog

nmn

m

yxn

m

n

mb

n

m

b

b

nbandmbthen

nyandmxLet

bbb

b

yx

y

x

yx

bb

logloglog

log

loglog

24

Evaluate: (no calculator)

20. 733log

21.

log3 3x 22.

9log3 5 23.

log3 94

Given log2 5 = m. Write the answer in terms of m.

24. Find log2 20 25. Find log225 = Given log12 9 = a & log12 18 = b. Write the answer in terms of a and b.

26. Find log12 (4

3) 27. Find log12216

Given log12 8 = k & log12 10 = p. Write the answer in terms of k and p.

28. Find log1280 29. Find log12 960

Since logarithmic functions and exponential functions are

inverses xax

a log and xaxa

log

You Always Know…

Loga ax = x Ex. Log3 35 = 5 because in exp. form

35 = 35

Loga a = 1 Ex. Log4 4 = 1 because in exp. form

41 = 4

Loga 1 = 0 Ex. Log7 1 = 0 because in exp. form

70 = 1

25

Day 9 CW: I. Express in expanded form using the properties of logs.

1. 𝑙𝑜𝑔37𝑥2 2. 𝑙𝑜𝑔48

√𝑥

3. 𝑙𝑜𝑔6√𝑥5

𝑦3 4. 𝑙𝑜𝑔5(2𝑎 ∙ 3𝑏)

II. Express as a single logarithm with coefficient of one. 5. 𝑙𝑜𝑔3𝑥 − 2𝑙𝑜𝑔3𝑦 6. 3𝑙𝑜𝑔3𝑎 + 4𝑙𝑜𝑔3𝑏 − 𝑙𝑜𝑔3𝑐 7. 𝑙𝑜𝑔2(𝑥 + 2) − 𝑙𝑜𝑔2(𝑥 − 1) 8. 𝑙𝑜𝑔(2𝑥 + 1) + 𝑙𝑜𝑔(3𝑥 − 2) 9. Given that 𝑙𝑜𝑔34 = 𝑥, 𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑒 𝑙𝑜𝑔316 𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑥.

10. If 𝑙𝑜𝑔𝑚𝑎 = 3 𝑎𝑛𝑑 𝑙𝑜𝑔𝑚𝑏 = 4, 𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑒 𝑙𝑜𝑔𝑚 (1

𝑎𝑏)

11. If log x = 3 evaluate log (10x2) 12. If log 6 2 = x and log 6 5 = y, express each logarithm in terms of x and y. a. log 6 40 b. log 6 3 Solve for x. 13. 12)(x

4log1)(2x

4log 14. 𝑙𝑜𝑔72𝑥 − 𝑙𝑜𝑔75 = 𝑙𝑜𝑔74

15. 2𝑙𝑜𝑔7𝑥 = 4𝑙𝑜𝑔72 + 2𝑙𝑜𝑔75 16. 𝑙𝑜𝑔3(𝑥 + 8) = 2 − 𝑙𝑜𝑔3𝑥

17. 𝑙𝑜𝑔2(𝑥2 − 2) − 𝑙𝑜𝑔2 (1

2𝑥 + 5) = 1 18. 𝑙𝑜𝑔25 + 2𝑙𝑜𝑔23 = 𝑙𝑜𝑔2𝑥 + 𝑙𝑜𝑔2(𝑥 − 4)

Laws of Exponents:

. number real any for,loglog

iffloglog

logloglog

logloglog

kMkM

NMNM

NMN

M

NMMN

bk

b

bb

bbb

bbb

26

Day 9 HW: NO CALCULATOR! Evaluate:

1. 813

log 2. 00010.log 3. 16

12

log 4. 525

log 5. 273

1log

6. 19

log 7. 48

log 8. 37log57log7

9. ))9(log(loglog 3210

Expand using the properties of Logs:

10. 27 3log x 11.

36logbc

a

Write as a single logarithm with a coefficient of 1.

12. cba 222 log5log3log 13. )log2log3(7

155 yx

Evaluate:

14. 53log23 15. 7log aa 16.

35log225log35 17.

122log62log22

18. 53log

9 19. 32log

8

Given: a3

2log and b5

2log find the following in terms of a and b:

20. 1802

log 21. 4502

.log 22. 25

62

log

Solve.

23. 216 x

log 24. 3

28

xlog 25. x255

log

26. 2

13

xlog 27. x8

16log 28. 2372

555logloglog x

29. 221 )log(log xx 30. 112 )log()log( xx 31. 416 xlog

32. 6

193

xlog 33. xx 56

4

2

4log)(log 34. 2

6xlog

35. x52.log 36.

4

13

xlog 37. 24

3 )(log x

38. )loglog(log 22426

1x 39. x

8

12

log 40. )(log)(log 51177

xx

27

Day 10 CW: Natural Log

Consider the following: y =

n

n

11

What happens as n approaches infinite? Let n = 10 y = n = 100 y = n = 1000 y = n = 10,000 y = n = 100,000 y = Round to three places: y = “e” _____________ “e” is the base for natural logs

Instead of using elog the symbol “ln” is used for log with base e.

Rewrite in Log or exp. form:

1. ln x = 5 2. 38972 .e 3. 12xe 4. ln 1 = 0 Simplify:

1. e-3 . e5 2. (3e4x)2 3.

32

4

1

e 4. xe24

5. ex . 4e2x+1 6. 3 964 xe

Solve: (round to 4 places) 1. e-x = 6 2. 1.2e-5x + 2.6 = 3 3. 0.25x - 0.5 = 2

4. 43

143

2xe

28

Day 10 CW/HW: I. Express in logarithmic form:

1. 102.33 = x 2. 10 x = 379.31 3. e2.5 = 12.18

4. 5

1

e = 1.221 5. e-2 = 0.135 6. 649.1e

II. Express in exponential form: 7. Log 229 = 2.3598 8. Log .8033 = -.0951 9. ln 8 = 2.079

10. 39.14

1ln 11. ln .01 = -4.605 12. ln e = 1

III. Express as a single Logarithm.

13. ln 6 + ln 5 – ln2 14. 6ln9ln2

1 15. 38ln

3

1 16. 24ln

2

3

IV. Find the Log of each number using a calculator. (Round to four decimal places) 17. Log 95 18. Log .233 19. Log 6.437 x 10-9 20. Log 1.8519 21. ln 120 22. ln 690

23. 5

6ln 24. 3

2

84ln 25. ln 0

V. Use a calculator to find the antilog ‘x’ of each Log or Natural Log. 26. Log x = 2.63 27. Log x = -.4089 28. Log x = 5.3 29. ln x = 2.208 30. ln x = 1.808 31. ln x = -.105

29

VI. Evaluate WITHOUT USING A CALCULATOR! 32. ln e 33. ln ( e2 ) 34. ln 1

35. ln 0 36. ln e

1 37. eln

38. ln (en ) 39. 7ln6ln e 40.

7ln2 e 41. 9ln

2

18ln

3

1

e

VII. Simplify and then Evaluate. (Use a calculator only when necessary)

42. ln 48 – 4 ln 2 43. 3ln212ln9ln2

1 44. 3ln25ln45ln

2

1

45. 6ln8ln e 46. ln 6 + ln 30 – (ln 5 + 3 ln 2) 47. )3ln2ln5(8ln4ln

2

1

48. 3 ln 4 – (ln 2 + ln 8 ) 49. )3(ln

2

1

e

VIII. Solve and Check! 50. Log (x + 3) + Log (x) = 1 51. Log (x + 3) - Log ( x – 1) = 1 52. Log x = 2 - Log (x + 21) IX. Solve by rewriting the problem in another form. Round decimal answers to four places!

53. ln x =3 54. 3

1ln x 55. 5xe

56. 5

3

ln ex 57. 12ln xe 58. 2

1ln x

30

Day 11 CW:

Change of base formula:

logb m logn m

logn b

Round all answers to 4 places.

To solve, PUT IN OTHER FORM. Or Take the log of both sides

1. log4 22 2. log12 95

3. 4x = 24 4. 7x = 20

5.

13x 5x2 6.

7x2 53x

Use logarithms to solve each equation. Round to 4 places. Do not use logs for #11,12 1. 3.5 x = 47.9 2. 8.2 y = 64.5 3. 7.2 a – 4 = 8.21 4. 2 b + 1 = 7.31

5. 5.78log3y 6. 8.91log 4k

7. 4 2x = 9 x – 1 8. 7 3b = 12 b + 2 9. 6 x – 2 = 4 3 – x 10. 3 4r = 5 r – 1

11. 4417 3

2

c 12. 1117 8

9

x

13. 30log7x 14. 152log3n

15. 725 32

x 16. xx 734 54

31

Day 11 HW: PRACTICE EXERCISES: Solve:

19) log (x+9) – log x = 1 20) log4(x+3) + log4(x-3) = 2 21) log5(x+4) +log5(x-4) = 2

22) xx loglog3

23) xx loglog4

24) 11log 25 x

25) 33 43 2 2logloglog xx

Key: 1) 3 6) –1 10) 1.1073 14) 3.5807 18) 1/3 22) 1, 109 2) 5 7) –3, -1 11) 2.7095 15) 5.6456 19) 1 23) 1, 1016

3) 3.3219 8) –3, 0.5 12) 7.4520 16) 10 20) 5 24) 2 6

4) 5.0444 9) 1.4037 13) 3.6067 17) 1 21) 41 25) 42

5) 5/2

1) 2x = 8 2) 2x = 32 3) 2x = 10 4) 2x = 33 5) 54x - 7 = 125 6) 43x + 5 = 16

7) 27

13 42

xx

8) 2793

25 xx

9) 4x = 7 10) 8x = 10 11) 2x = 3x - 1 12) 3x + 2 = 5x - 1 13) (2.8)x = 41 14) (3.4)x = 80 15) (1.7)x = 20 16) log x + log (x–9) = 1 17) log x + log (x+9) = 1 18) log x – log (x+3) = -1

32

Day 12 CW: FORMULAS:

Compound Interest :

tn

n

rPA

1 Continuously Compounded :

trPeA

“n times a year “

Value of an Asset : trPV 1 Growth &/ Decay : tkney

CLASS EXAMPLES: 1.) How long would it take for an investment of $2500 to triple if it is invested in an account that earns 6% interest compounded quarterly.

2.) Your bank promises to double your money in 2

18 years. Assuming the interest rate is compounded

continuously, what is the interest rate? 3.) The half-life of a radioactive isotope is 9 years. Find the constant “k” for a 20 gram sample. 4.) Zeller industries bought some equipment for $50,000. It is expected to depreciate at a steady rate of 10% a year. When will the value be half the original value? 5.) The Jamesons bought s new house for $144,500 five years ago. The home is now worth $187,850. Assuming a steady rate of growth , what was the yearly rate of appreciation?

33

Day 12 HW: Applications of Logarithms Solve each problem. Show all work! 1. 2. 3. 4. 5. 6. 7. 8.

Suppose $500 is invested at 6% annual interest compounded twice a year. When will the investment be worth $1000?

Suppose $500 is invested at 6% annual interest compounded continuously. When will the investment be worth $1000?

An organism of a certain type can grow from 30 to 195 organisms in 5 hours. Find k for the growth formula.

For a certain strain of bacteria, k is 0.825 when t is measured in days. How long will it take 20 bacteria to increase to 2000?

An investment service promises to triple your money in 12 years. Assuming continuous compounding of interest, what rate of interest is needed?

A substance decomposes radioactively. Its half-life is 32 years. Find the constant k in the decay formula.

A piece of machinery valued at $250,000 depreciates at 12% per year by the fixed rate method. After how many years will the value have depreciated to $100,000?

Mike bought a new car 8 years ago for $5400. To buy a new car comparably equipped now would cost $12,500. Assuming a steady rate of increase, what was the yearly rate of inflation in car prices over the 8-year period?

34

Review Sheet. NO CALC, unless rounding is stated *1. Write an equivalent logarithmic equation for x5 = 32. 1. _______________________

*2. Write an equivalent exponential equation for 5ln x 2. ______________________

*3. What is the domain and range of f(x) = ex + 2? 3. D: _____________ R: ____________ 4. The log 0.034 is between what two consecutive integers? 4. ______________________ 5. Given log 8.1 = a, find the log 8100 in terms of a. 5. ______________________ *6. Given f(x) = 3x and g(x) = x9log , find f-1( g (9)). 6. ______________________

7. Write as a single logarithm. 2

ln1 x 7. ______________________

Simplify Completely:

*8. 64

1log8 9. ln e9 *10. 2

3

2 4log *11. 4log8 12. 5

3log

27

Solve the following: *13. 01log x *14. x100log1000 *15. x354 log)25(loglog

*16. 181824 x *17. 0))(log(loglog 32 x

18. 2

ln54 xex *19. 68log x

*20. x7

8

1 16log *21. 3

2

25

1log x

*22. 7log)3(loglog 92 x 23. 2)2(loglog2 33 xx

35

24. )5log2(log220loglog2

17777 x

25. )6log3log2(log2

3)5log215(log2log 33 x

26. If a5log find log 0.04 in terms of a.

27. If 2log3c and 5log3d , express 5

104log 3 in terms of c and d.

Solve. Round to 4 places.

28. 131 5.46 xx 29. 5.74)1(2 3

4

x

30. 0103732 xx 31. 3.35.21

44

xe

36

Decay/Growth Value of an Asset Compounded continuously Compounded “n” times/year

y = nekt V = P ( 1 r)t A = Pert tn

n

rPA

1

32. To the nearest dollar, what amount must be invested at 6% compounded continuously for 14 years in order for a balance to be $23,140? 33. A tractor that 4 years ago cost $8,000, now is worth only $3200. Find the average annual rate of depreciation. 34. The population of a certain colony of bacteria doubles every 5 hours. How long will it take for the population to triple? 35. Radioactive iodine is used to determine the health of the thyroid gland. It decays according the formula y = ne -.0856t where t is in days. Find the *half-life of this substance. (*time it takes for a substance to be half its original mass)

36. A radioisotope is used as a power source for a satellite. The power output is given by 25050

t

eP

where P is the power in watts and t is time in days. a) Find the power available after 100 days. b) If ten watts of power are required to operate, how long can the satellite continue to operate? Graph on graph paper: 37. y = 2x-2 -1 38. yx

2

1log

Solve for x.

39. xx 55 40. 15 xx 41. 2310 xx

37

Extra Review Use ONLY a (4 function) calculator . Write in logarithmic form.

1. 72 = 49 2. x6

1

5 Write in exponential form.

3. 2819

log 4. 532

12

log

Solve for x.

5. 34

xlog 6. 481xlog 7. 3

227

xlog 8. 28

xlog

Write in expanded form.

9. 6

2

5

xlog 10.

3

45xylog 11.

5 4axlog

Write as a single logarithm with a coefficient of 1.

12. 632777

logloglog xy 13. 5123bbb

xy log)(loglog

Solve for x.

14. 732999

logloglog x 15. 392666

logloglog x

16. 2644

)(loglog xx 17. 382

2 )(log x

If log12 8 = a and log12 10 = b, find the following in terms of a and/or b:

18. 80012

log 19. 812

log

Solve for x: (give the calculator ready answer)

20.

5.3x 72