what’s the inverse of an exponent? logarithmic functions...

Chapter 7 l Skills Practice 569

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Skills Practice Skills Practice for Lesson 7.1

Name _____________________________________________ Date ____________________

What’s the Inverse of an Exponent?Logarithmic Functions as Inverses

Vocabulary

Write the term that best completes each statement.

1. The of a number to a given base is the power or

exponent to which the base must be raised in order to produce the number.

2. A(n) is a function involving a logarithm.

3. A(n) is a logarithm with base 10.

4. A(n) is a logarithm with base e.

570 Chapter 7 l Skills Practice

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Problem Set

Graph each exponential function f(x) and the line y � x, and label each graph. Complete the tables for each exponential function and its inverse. Then plot each point on the grid provided. Connect the points of f�1(x) with a smooth curve. Then label the graph as f�1(x).

1. f(x) � 4x

x f(x) � 4x x f�1(x)

�2 1 ___ 16

1 ___ 16

�2

�1 1 __ 4

1 __ 4 �1

0 1 1 0

1 4 4 1

2 16 16 2

f(x) = 4X

y = x

f –1(x)

x1612

4

8

12

16

–4–4

840–8

–8

–12

–12

–16

–16

y


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2. f(x) � 5x

x f(x) � 5x x f�1(x)

�2

�1

0

1

2


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3. f(x) � 6x

x f(x) � 6x x f�1(x)

�2

�1

0

1

2


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4. f(x) � ( 1 __ 2 )

x

x f(x) � ( 1 __ 2 ) x x f�1(x)

�3

�2

�1

0

1

2

3


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5. f(x) � ( 1 __ 3 )

x

x f(x) � ( 1 __ 3

) x x f�1(x)

�3

�2

�1

0

1

2

3


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6. f(x) � ( 1 __ 4 )

x

x f(x) � ( 1 __ 4 ) x x f�1(x)

�2

�1

0

1

2


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The graph of the function h(x) � b x and the line y � x are shown.

Sketch the graph of h�1(x).

7.

x

h–1(x)

y = xh(x)

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y 8.

x

y = x

h(x)

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

9.

x

y = x

h(x)

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y 10.

x

y = xh(x)

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

11.

x

y = xh(x)

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y 12.

x

y = x

h(x)

86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y


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Name _____________________________________________ Date ____________________

Do I Have the Right Form?Exponential and Logarithmic Forms

Problem Set

Write each exponential equation as a logarithmic equation using the definition of logarithms.

1. 32 � 9 2. 43 � 64

log3 9 � 2

3. 103 � 1000 4. 82 � 64

5. 25 � 32 6. ( 1 __ 2 )

4

� 1 ___ 16

7. ( 1 __ 3 )

�4

� 81 8. 4�2 � 1 ___ 16

9. ( 1 __ 6 )

�3

� 216 10. 5�3 � 1 ____ 125

11. 16 1 __

2 � 4 12. 1000

1 __

3 � 10

13. ( 3 __ 5 )

5

� 243

_____ 3125

14. 9 3 __ 2 � 27

15. 4 5 __

2 � 32 16. 1000

4 __

3 � 10,000


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Write each logarithmic equation as an exponential equation using the definition of logarithms.

17. log3 81 � 4 18. log

5 125 � 3

34 � 81

19. log 1 __ 3

81 � �4 20. log

3 1 __ 9 � �2

21. log10

0.001 � �3 22. log 1 __ 2 8 � �3

23. log6 36 � 2 24. log

4 1 ___ 64

� �3

25. log 3 __ 4

243 _____ 1024

� 5 26. l og 2 __ 5

16 ____ 625

� 4

27. log 2 __ 3

81 ___ 16

� �4 28. log10

0.0001 � �4

Evaluate each logarithmic expression without using a calculator. Explain how you calculated each.

29. log3 9 30. log

4 64

32 � 9, so log3 9 � 2

31. log 1 __ 2

8 32. log 10,000

33. log16

4 34. log 1 __ 4

1 __ 8

35. log8 3 �� 6 4 36. log

5 �� 100,000


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Evaluate each logarithmic expression. Use a calculator if necessary.

37. log 1000 38. log 68

3

39. log 0.09 40. log 0.0001

41. log 400 42. log 1 __

8

43. log 54.5 44. log 4 �� 100,00 0

45. ln 1000 46. ln 42

47. ln 0.4 48. In e 1 __ 2

49. ln 0.0025 50. ln 3 �� 100,00 0

51. ln 8 52. ln √___

50

53. 2 log 10 54. 3 log 8

55. log 0.97 56. In e 5 __ 2

57. In 3 __ 5 58. log

4 �� 8 1

59. ln 9.28 60. 1 __ 2 In √

___

30


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Name _____________________________________________ Date ____________________

It’s All in the GraphGraphs of Logarithmic Functions

Problem Set

Graph each function f(x) for y-values between 0 and 100 and then again for y-values between 0 and 1000. Describe the similarities and differences between the graphs.

1. f(x) � 2x

x

–2–4–6–8

y

30

40

50

60

70

80

90

100

20

10

8642

100

200

300

400

500

y

600

700

800

900

1000

x

86–2 42–4–6–8

Each graph has an asymptote at y � 0 and increases very rapidly.

2. f(x) � 3x


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3. f(x) � 4x

4. f(x) � ( 1 __ 2 )

x


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5. f(x) � ( 1 __ 3 )

x

6. f(x) � ex


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Graph each logarithmic function. Label the coordinates of three points. Give the domain, range, intercepts, and asymptotes of the graph. Then give the x-values for which the function is increasing or decreasing.

7. f(x) � log x

x(1, 0)

(10, 1)

( )110

, –187 93 65421

1

2

3

4

–1

–2

–3

–4

y Domain: all non-negative real numbers Range: all real numbers Intercept(s): x-intercept at x � 1, no y-intercepts Asymptote(s): vertical asymptote at the y-axis, or x � 0 x-values for which the function is

increasing or decreasing: increasing over the entire domain

8. f(x) � log2 x

Domain:

Range:

Intercept(s):

Asymptote(s):

x-values for which the function is

increasing or decreasing:


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9. f(x) � log3 x

Domain:

Range:

Intercept(s):

Asymptote(s):



10. f(x) � log4

x

Domain:

Range:

Intercept(s):

Asymptote(s):




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11. f(x) � In x

Domain:

Range:

Intercept(s):

Asymptote(s):



12. f(x) � log 1 __

2 x

Domain:

Range:

Intercept(s):

Asymptote(s):




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13. f(x) � log 1 __

3 x

Domain:

Range:

Intercept(s):

Asymptote(s):



14. f(x) � log 1 __

4 x

Domain:

Range:

Intercept(s): Asymptote(s):




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Name _____________________________________________ Date ____________________

Transformers Again!Transformations of Logarithmic Functions

Problem Set

Sketch and label the graph of the first function in each group. Then use that graph to sketch and label the graphs of the other two functions.

1. f(x) � log x; f(x) � log

x � 2; f(x) � log

x � 3

x

f(x) = log x + 2

f(x) = log x

f(x) = log x – 3

1614 186 1210842

1

2

3

4

–1

2

–3

–4

y

2. f(x) � log x; f(x) � log

(x � 3); f(x) � log (x � 2)


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3. f(x) � log x; f(x) � log x � 3; f(x) � log x � 1

4. f(x) � In x; f(x) � In

(x � 3); f(x) � In (x � 2)


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5. f(x) � In x; f(x) � In (x � 1) � 2; f(x) � In (x � 2) � 2

6. f(x) � log x; f(x) � log (x � 1) � 3; f(x) � log (x � 2) � 2


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Sketch and label the graphs of each function.

7. f(x) � log x and f(x) � log (�x)

x

f(x) = log xf(x) = log(–x)

129

1

2

3

4

–3–1

63–6

–2

–9

–3

–12

–4

y

8. f(x) � log x and f(x) � �log x


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9. f(x) � log x; f(x) � 2 log x; f(x) � 2 log (�x)

10. f(x) � In x; f(x) � �2 In x; f(x) � �2 In (�x)


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11. f(x) � log x; f(x) � 1 __

2 log (�x); f(x) � � 1 __

2 log x

12. f(x) � log x; f(x) � log ( � 1 __ 2 x ) ; f(x) � �log ( � 1 __

2 x )


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13. f(x) � In x; f(x) � �2 In ( 1 __

2 x ) ; f(x) � �2 In ( � 1 __

2 x )

14. f(x) � log x; f(x) � log ( 3 __

2 x ) ; f(x) � log ( � 3 __

2 x )


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15. f(x) � log x and f(x) � 2 log x � 1

16. f(x) � log x and f(x) � �log(x � 2) � 1


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17. f(x) � log x and f(x) � 2 log (�(x � 2))

18. f(x) � In x and f(x) � In (�x) � 3


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19. f(x) � log x and f(x) � log ( � 1 __ 3 ( x � 3) )

20. f(x) � In x and f(x) � � 1 __ 2 In ( x � 2)


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21. f(x) � log x and f(x) � �log ( 1 __

2 x ) � 3

22. f(x) � In x and f(x) � �3 In (2 x) � 1

what’s the inverse of an exponent? logarithmic functions...

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