Chapter 10 Esponential and Logarithmic Relations Lesson 10-1 ...

[Pages:22]Chapter 10 Esponential and Logarithmic Relations Lesson 10-1 Exponential Functions Pages 527?530

1. Sample answer: 0.8

3. c 5. b

2a. quadratic 2b. exponential 2c. linear 2d. exponential

4. a

6. D {x 0 x is all real numbers.}, R {y 0 y 0}

y

7. D {x 0 x is all real numbers.}, R {y 0 y 0}

y

y 3(4)x

O

x

8. growth

( ) y 2

1 3

x

O

x

9. decay 11. y 3a12bx 13. 2227 or 427 15. 3322 or 2722 17. x 0 19. y 65,000(6.20)x

10. growth

12. y 18132x

14. a4 16. 9 18. 2 20. 22,890,495,000

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Algebra 2 Chapter 10

21. D {x 0 x is all real numbers.}, R {y 0 y 0}

y

22. D {x 0 x is all real numbers.}, R {y 0 y 0}

y

y 2(3)x

O

x

y 5(2)x

O

x

23. D {x 0 x is all real numbers.}, R {y 0 y 0}

y

y 0.5(4)x

24. D {x 0 x is all real numbers.}, R {y 0 y 0}

y

( ) y 4

1 3

x

O

x

25. D {x 0 x is all real numbers.}, R {y 0 y 0}

y x

O

( ) y

1 5

x

O

x

26. D {x 0 x is all real numbers.}, R {y 0y 0}

y

O

x

y 2.5(5)x

27. growth

29. decay

31. decay

33.

y

2

1x a4b

35. y 7(3)x

37. y 0.2(4)x

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28. growth 30. growth 32. decay

34. y 3(5)x

36.

y

5

1x a3b

38. y 0.3(2)x

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Algebra 2 Chapter 10

39. 54 or 625 41. 7412 43. n 2

45. n 5

47. 1

49.

8

3

51. n 3

53. 3 55. 10 57. y 100(6.32)x 59. y 3.93(1.35)x

61. 2144.97 million; 281.42 million; No, the growth rate has slowed considerably. The population in 2000 was much smaller than the equation predicts it would be.

63. A(t ) 1000(1.01)4t

65. s . 4x

67. Sometimes; true when b 1, but false when b 1.

40. x115 42. y 2 13

44. 25

46.

2 3

48. n 2

50. 0

52.

5 3

54. p 2

56. 3, 5

58. about 1,008,290

60. 9.67 million; 17.62 million;

32.12 million; These answers

are in close agreement with

the actual populations in

those years.

62.

Exponential; the base, 1 r ,

n

is fixed, but the exponent, nt,

is variable since the time t

can vary.

64. $2216.72

66. 1.5 three-year periods or 4.5 yr

68. The number of teams y that could compete in a tournament with x rounds can be expressed as y 2x. The 2 teams that make it to the final round got there as a result of winning games played with 2 other teams, for a total of 2 2 22 or 4 games played in the previous rounds. Answers should include the following.

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69. A 71.

? Rewrite 128 as a power of 2, 27. Substitute 27 for y in the equation y 2x. Then, using the Property of Equality for Exponents, x must be 7. Therefore, 128 teams would need to play 7 rounds of tournament play.

? Sample answer: 52 would be an inappropriate number of teams to play in this type of tournament because 52 is not a power of 2.

70. 780.25

72.

[5, 5] scl: 1 by [1, 9] scl: 1

The graphs have the same

shape. The graph of y 2x 3 is the graph of y 2x translated three units

up. The asymptote for the graph of y 2x is the line y 0 and for y 2x 3 is

the line y 3. The graphs

have the same domain, all

real numbers, but the range of y 2x is y 0 and the range of y 2x 3 is y 3.

The y-intercept of the graph of y 2x is 1 and for the graph of y 2x 3 is 4.

[5, 5] scl: 1 by [1, 9] scl: 1

The graphs have the same

shape. The graph of y 3x1 is the graph of y 3x translated one unit to

the left. The asymptote for the graph of y 3x and for y 3x1 is the line y 0.

The graphs have the same

domain, all real numbers,

and range, y 0. The

y-intercept of the graph of y 3x is 1 and for the graph of y 3x1 is 3.

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Algebra 2 Chapter 10

73.

74.

[5, 5] scl: 1 by [1, 9] scl: 1

The graphs have the same shape. The graph of

y

1 x2 a5b

is

the

graph

of

y

1x a5b

translated

two

units

to the right. The asymptote

for

the

graph

of

y

1x a5b

and

for

y

1 x2 a5b

is

the

line

y 0. The graphs have the same domain, all real numbers, and range, y 0. The y-intercept of the graph

of

y

1x a5b

is

1

and

for

the

graph

of

y

1 x2 a5b

is

25.

75. For h 0, the graph of y 2x is translated 0h 0 units

to the right. For h 0, the graph of y 2x is translated

|h| units to the left. For k 0, the graph of y 2x is

translated 0k 0 units up. For k 0, the graph of y 2x is

translated 0k 0 units down.

[5, 5] scl: 1 by [3, 7] scl: 1

The graphs have the same shape. The graph of

y

a

1 4

x

b

1

is

the

graph

of

y

1x a4b

translated

one

unit down. The asymptote

for

the

graph

of

y

1x a4b

is

the line y 0 and

for

the

graph

of

y

1x a4b

1

is the line y 1. The graphs have the same domain, all real numbers,

but

the

range

of

y

1x a4b

is

y

0

and

of

y

1x a4b

1

is y 1. The y-intercept

of

the

graph

of

y

1x a4b

is

1

and for the graph of

y

1x a4b

1

is

0.

76. 1, 15

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Algebra 2 Chapter 10

77. 1, 6 79. 0 x 3 or x 6

78. 133, 3 80. square root

81. greatest integer

83. B10 01R

85.

1 51

3 B11

6 5R

87. g [h(x)] 2x 6; h [g(x)] 2x 11

89. g [h(x)] 2x 2; h [g(x)] 2x 11

82. constant

y

y 8

O

x

84. does not exist

86. about 23.94 cm

88. g [h(x)] x 2 6x 9; h [g(x)] x 2 3

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Algebra 2 Chapter 10

Lesson 10-2 Logarithms and Logarithmic Functions Pages 535?538

1. Sample answer: x 5y and y log5 x

3. Scott; the value of a logarithmic equation, 9, is the exponent of the equivalent exponential equation, and the base of the logarithmic expression, 3, is the base of the exponential equation. Thus x 39 or 19,683.

5.

log7

1 49

2

7. 3621 6

9. 3

11. 1

13. 1000

15. 12, 1

17. 3 19. 107.5

21. log8 512 3

23.

log5

1 125

3

25.

log100

10

1 2

27. 53 125

29.

41

1 4

31.

82 3

4

33. 4

2. They are inverses. 4. log5 625 4

6. 34 81

8. 4

10. 21

12. 27

14.

1 2

x

5

16. x 6

18. 1013 20. 105.5 or about 316,228 times 22. log3 27 3

24. log1 9 2 3

26.

log2401

7

1 4

28. 132 169

30.

10012

1 10

32.

1 2 a5b

25

34. 2

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Algebra 2 Chapter 10

35.

1 2

37. 5

39. 7

41. n 5

43. 3

45. 1018.8

47. 81

49. 0 y 8

51. 7

53. x 24

55. 4

57. 2

59. 5

61. a 3

63. log5 25 ? 2 log5 5 Original

equation

log5 52 ? 2 log5 51 2 ? 2(1)

25 52 and 5 51

Inverse Prop. of Exp. and Logarithms

2 2

Simplify.

36.

5 2

38. 4

40. 45

42. 3x 2

44. 2x

46. 1010.67

48. c 256

50. 125

52. 0 p 1

54. 3

56. 11

58. 25

60. y 3

62. 8

64.

log16 2 log2 16 ? 1

1

log16 164

log2 24

? 1

Original equation

2 1641 and 16 24

1 4

(4)

?

1

Inverse Prop. of Exp. and Logarithms

1 1

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