Upload

hadiep

View

244

Download

0

Embed Size (px)

Chapter 9

A7

Glencoe A

lgebra 2

Answers

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

An

swers

(Lesson 9-2)

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

9-2NAME ______________________________________________ DATE______________ PERIOD _____

Chapter 9 17 Glencoe Algebra 2

Less

on

9-2

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 An equation for loudness, in decibels, is L � 10 log10 R, where R is the relativeintensity of the sound. Sounds that reach levels of 120 decibels or more are painful tohumans. What is the relative intensity of 120 decibels? 1012

38. INVESTING Maria invests $1000 in a savings account that pays 4% interestcompounded annually. The value of the account A at the end of five years can bedetermined from the equation log A � log[1000(1 � 0.04)5]. Find the value of A to thenearest dollar. $1217

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

PracticeLogarithms and Logarithmic Functions

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

9-2

Chapter 9 16 Glencoe Algebra 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

NAME ______________________________________________ DATE______________ PERIOD _____

9-29-2 Skills PracticeLogarithms and Logarithmic Functions

Chapter 9

A10

Glencoe A

lgebra 2

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

An

swers

(Lesson 9-3)

Skills PracticeProperties of Logarithms

9-3NAME ______________________________________________ DATE______________ PERIOD _____

Chapter 9 23 Glencoe Algebra 2

Less

on

9-3

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

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

9-3

Chapter 9 22 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 _____

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

9-3

Exercises

Example

Chapter 9

A11

Glencoe A

lgebra 2

Answers

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

An

swers

(Lesson 9-3)

9-3 Word Problem PracticeProperties of Logarithms

NAME ______________________________________________ DATE______________ PERIOD _____

Chapter 9 25 Glencoe Algebra 2

Less

on

9-3

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

1. MENTAL COMPUTATION Jessica hasmemorized log5 2 � 0.4307 and log5 3 �0.6826. Using this information, to thenearest thousandth, what power of 5 isequal to 6?1.113

2. POWERS A chemist is formulating anacid. The pH of a solution is given by

�log10 C,

where C is the concentration ofhydrogen ions. If the concentration ofhydrogen ions is increased by a factor of 100, what happens to the pH of thesolution?The pH decreases by 2.

3. LUCKY MATH Frank is solving aproblem involving logarithms. He doeseverything correctly except for one thing.He mistakenly writes

log2 a � log2 b � log2 (a � b).

However, after substituting the valuesfor a and b in his problem, he amazinglystill gets the right answer! The value of a was 11. What must the value of bhave been?1.1

4. LENGTHS Charles has two poles. Onepole has length equal to log7 21 and the other has length equal to log7 25.Express the length of both poles joinedend to end as the logarithm of a singlenumber.log7 525

SIZE For Exercises 5-7, use thefollowing information.

Alicia wanted to try to quantify the termspuny, tiny, small, medium, large, big, huge,and humongous. She picked a number of objects and classified them with theseadjectives of size. She noticed that the scaleseemed exponential. Therefore, she came upwith the following definition. Define S to be

log3 V, where V is volume in cubic feet.

Then use the following table to find theappropriate adjective.

5. Derive an expression for S applied to a cube in terms of � where � is the sidelength of a cube.log3 �

6. How many cubes, each one foot on a side, would have to be put together to get an object that Alicia would call“big”?729

7. How likely is it that a large objectattached to a big object would result in a huge object, according to Alicia’s scale.Not very likely; most likely theresult will be big, not huge.

S satisfies Adjective

�2 S � �1 tiny

�1 S � 0 small

0 S � 1 medium

1 S � 2 large

2 S � 3 big

3 S � 4 huge

1�3

PracticeProperties of Logarithms

9-3

Chapter 9 24 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 Recall that the loudness L of a sound in decibels is given by L � 10 log10 R,where R is the sound’s relative intensity. If the intensity of a certain sound is tripled, byhow many decibels 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

3�2

2�3

25�7

5�7

7�5

NAME ______________________________________________ DATE______________ PERIOD _____

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

9-3