Download Precalculus: A Prelude to Calculus 1st Edition Paper

section 3.2 Logarithms as Inverses of Exponentiation 243
Evaluate log2 73.9.
example 2
solution Use a calculator with b = 2 and y = 73.9 in the formula above, getting
log 73.9
≈ 6.2075.
log 2
log2 73.9 =
The change of base formula for logarithms implies that the graph of the
logarithm using any base can be obtained from vertically stretching the graph
of the logarithm using any other base, as shown in the following example.
Sketch the graphs of log2 x and log x on the interval [ 18 , 8]. What is the relationship
between these two graphs?
example 3
solution The change of base formula implies that log x = (log 2)(log2 x). Because
log 2 ≈ 0.3, this means that the graph of log x is obtained from the graph of log2 x
(sketched earlier) by stretching vertically by a factor of approximately 0.3.
y
3
2
1
1
2
3
4
5
6
7
8
x
The graphs of log2 x (blue)
and log x (red) on the
1
interval [ 8 , 8].
1
2
3
exercises
For Exercises 1–16, evaluate the indicated expression. Do not use a calculator for these exercises.
1. log2 64
2. log2 1024
1
128
1
log2 256
3. log2
4.
9. log 10000
10. log
1
1000
√
11. log 1000
12.
1
log √10000
log2 83.1
17. Find a number y such that log2 y = 7.
18. Find a number t such that log2 t = 8.
19. Find a number y such that log2 y = −5.
20. Find a number t such that log2 t = −9.
For Exercises 21–28, find a number b such that
the indicated equality holds.
21. logb 64 = 1
25. logb 64 = 12
6.3
14. log8 2
22. logb 64 = 2
26. logb 64 = 18
7. log4 8
15. log16 32
23. logb 64 = 3
27. logb 64 =
8. log8 128
16. log27 81
24. logb 64 = 6
28. logb 64 =
5. log4 2
6. log8 2
13.
3
2
6
5
244
chapter 3 Exponents and Logarithms
29. Find a number x such that log3 (5x + 1) = 2.
30. Find a number x such that log4 (3x + 1) = −2.
For Exercises 55–58, find a formula for (f ◦ g)(x)
assuming that f and g are the indicated
functions.
31.
Find a number x such that 13 = 102x .
32.
Find a number x such that 59 = 103x .
55. f (x) = log6 x
and
g(x) = 63x
33.
Find a number t such that
56. f (x) = log5 x
and
g(x) = 53+2x
10 + 1
= 0.8.
10t + 2
t
34.
57. f (x) = 63x
58. f (x) = 5
Find a number t such that
10
36.
60. Find a number n such that log3 (log2 n) = 2.
61. Find a number m such that log7 (log8 m) = 2.
Find a number x such that
2x
and g(x) = log5 x
59. Find a number n such that log3 (log5 n) = 1.
10t + 3.8
= 1.1.
10t + 3
35.
and g(x) = log6 x
3+2x
x
+ 10 = 12.
62. Find a number m such that log5 (log6 m) = 3.
Find a number x such that
For Exercises 37–54, find a formula for the inverse function f −1 of the indicated function f .
For Exercises 63–70, evaluate the indicated quantities. Your calculator is unlikely to be able to
evaluate logarithms using any of the bases in
these exercises, so you will need to use an appropriate change of base formula.
37. f (x) = 3x
47. f (x) = log8 x
63.
log2 13
67.
log9 0.23
38. f (x) = 4.7
48. f (x) = log3 x
64.
log4 27
68.
log7 0.58
39. f (x) = 2
49. f (x) = log4 (3x + 1)
65.
log13 9.72
69.
log4.38 7.1
40. f (x) = 9
50. f (x) = log7 (2x − 9)
66.
log17 12.31
70.
log5.06 99.2
41. f (x) = 6x + 7
51. f (x) =
5 + 3 log6 (2x + 1)
102x − 3 · 10x = 18.
x
x−5
x+6
42. f (x) = 5x − 3
43. f (x) = 4 · 5x
44. f (x) = 8 · 7
x
52. f (x) =
8 + 9 log2 (4x − 7)
45. f (x) = 2 · 9x + 1
53. f (x) = logx 13
46. f (x) = 3 · 4 − 5
54. f (x) = log5x 6
x
problems
71. Explain why log3 100 is between 4 and 5.
72. Explain why log40 3 is between
1
4
and
1
.
3
73. Show that log2 3 is an irrational number.
[Hint: Use proof by contradiction: Assume that
m
log2 3 is equal to a rational number n ; write
out what this means, and think about even and
odd numbers.]
74. Show that log 2 is irrational.
75. Explain why logarithms with base 0 are not defined.
76. Explain why logarithms with a negative base are
not defined.
77. Explain why log5
√
5 = 12 .
78. Suppose a and b are positive numbers, with
a = 1 and b = 1. Show that
loga b =
1
.
logb a
79. Suppose b and y are positive numbers, with
1
b = 1 and b = 2 . Show that
log2b y =
logb y
.
1 + logb 2