Download SECTION 5-3 Logarithmic Functions

376
★
5 Exponential and Logarithmic Functions
64. Medicine. An electronic heart pacemaker utilizes the same
type of circuit as the flash unit in Problem 63, but it is designed so that the capacitor discharges 72 times a minute.
For a particular pacemaker, the charge on the capacitor t
seconds after it starts recharging is given by
company to learn to test a particular model of a personal
computer after it comes off the assembly line. The learning
curve for an average trainee is given by
N
q 0.000 008(1 e2t)
where N is the number of computers tested per day after t
days on the job. Graph the equation for 0 t 50. What is
the maximum number of computers that an average tester
can be expected to test per day after training?
Graph this function for 0 t 56. Estimate the maximum
charge on the capacitor.
★
65. Wildlife Management. A herd of 20 white-tailed deer is
introduced to a coastal island where there had been no deer
before. Their population is predicted to increase according
to the logistic curve
N
★
100
1 4e0.14t
67. Catenary. A free-hanging power line between two supporting towers looks something like a parabola, but it is actually a curve called a catenary. A catenary has an equation
of the form
y
where N is the number of deer expected in the herd after t
years. Graph the equation for 0 t 30. Estimate the herd
size that the island can support.
★
66. Training. A trainee is hired by a computer manufacturing
SECTION
5-3
200
4 21e0.1t
emx emx
2m
where m is a constant. Graph the equation for m 0.25.
★
68. Catenary. Graph the equation in Problem 67 for m 0.4.
Logarithmic Functions
• Definition of Logarithmic Function
• From Logarithmic Form to Exponential Form, and Vice Versa
• Properties of Logarithmic Functions
• Definition of
Logarithmic Function
We now define a new class of functions, called logarithmic functions, as inverses of
exponential functions. Since exponential functions are one-to-one, their inverses exist.
Here you will see why we placed special emphasis on the general concept of inverse
functions in Section 3-6. If you know quite a bit about a function, then, based on
a knowledge of inverses in general, you will automatically know quite a bit about
its inverse. For example, the graph of f 1 is the graph of f reflected across the line
y x, and the domain and range of f 1 are, respectively, the range and domain of f.
If we start with the exponential function,
f:
y 2x
and interchange the variables x and y, we obtain the inverse of f:
f 1: x 2 y
The graphs of f, f 1, and the line y x are shown in Figure 1. This new function is
given the name logarithmic function with base 2. Since we cannot solve the equation x 2 y for y using the algebra properties discussed so far, we introduce a new
symbol to represent this inverse function:
y log2 x
Read “log to the base 2 of x.”
377
5-3 Logarithmic Functions
Thus,
y log2 x
x 2y
is equivalent to
that is, log2 x is the exponent to which 2 must be raised to obtain x. Symbolically,
x 2 y 2log x.
2
FIGURE 1 Logarithmic function
with base 2.
y
f
y 2x
10
f 1
x 2y
or
y log2 x
5
5
5
10
x
5
DOMAIN of f (, ) RANGE of f
RANGE of f (0, ) DOMAIN of f 1
f 1
f
yx
x
y 2x
x 2y
y
3
1
8
1
8
3
2
1
4
1
4
2
1
1
2
1
2
1
0
1
1
0
1
2
2
1
2
4
4
2
3
8
8
3
1
Ordered
pairs
reversed
In general, we define the logarithmic function with base b to be the inverse of
the exponential function with base b (b 0, b 1).
DEFINITION 1
Definition of Logarithmic Function
For b 0 and b 1,
Logarithmic form
y logb x
Exponential form
is equivalent to
x by
The log to the base b of x is the exponent to which b must be raised to
obtain x.
y log10 x
is equivalent to
x 10y
y loge x
is equivalent to
x ey
Remember: A logarithm is an exponent.
It is very important to remember that y log b x and x b y define the same
function, and as such can be used interchangeably.
378
5 Exponential and Logarithmic Functions
Since the domain of an exponential function includes all real numbers and its
range is the set of positive real numbers, the domain of a logarithmic function is the
set of all positive real numbers and its range is the set of all real numbers. Thus,
log10 3 is defined, but log10 0 and log10 (5) are not defined. That is, 3 is a logarithmic domain value, but 0 and 5 are not. Typical logarithmic curves are shown in
Figure 2.
FIGURE 2 Typical logarithmic
graphs.
y
y
y logb x
b1
y logb x
0b1
0
x
1
DOMAIN (0, )
RANGE (, )
0
DOMAIN (0, )
RANGE (, )
(a)
EXPLORE-DISCUSS 1
• From Logarithmic
Form to Exponential
Form, and Vice Versa
EXAMPLE 1
x
1
(b)
For the exponential function f {(x, y) y (23)x}, graph f, f 1, and y x on the
same coordinate system. Discuss the domains and ranges of f and its inverse. By
what other name is f 1 known?
We now look into the matter of converting logarithmic forms to equivalent exponential forms, and vice versa.
Logarithmic–Exponential Conversions
Change each logarithmic form to an equivalent exponential form:
(A) log2 8 3
Solutions
Matched Problem 1
(A) log2 8 3
(B) log25 5 12
(C) log2 ( 14) 2
(C) log2 ( 14) 2
1
2
8 23
5 251/2
1
2
4 2
is equivalent to
is equivalent to
is equivalent to
Change each logarithmic form to an equivalent exponential form:
(A) log3 27 3
EXAMPLE 2
(B) log25 5 (B) log36 6 (C) log3 ( 19) 2
1
2
–Exponential Conversions
Logarithmic–
Change each exponential form to an equivalent logarithmic form:
(A) 49 72
(B) 3 9
(C)
1
5
51
5-3 Logarithmic Functions
Solutions
Matched Problem 2
(A) 49 72
(B) 3 9
(C) 15 51
is equivalent to
is equivalent to
is equivalent to
379
log7 49 2
log9 3 21
log5 ( 15) 1
Change each exponential form to an equivalent logarithmic form:
(A) 64 43
3
(B) 2 8
(C)
1
16
42
To gain a little deeper understanding of logarithmic functions and their relationship to the exponential functions, we consider a few problems where we want to find
x, b, or y in y logb x, given the other two values. All values were chosen so that
the problems can be solved without tables or a calculator.
EXAMPLE 3
Solutions of the Equation y logb x
Find x, b, or y as indicated:
(A) Find y: y log4 8
(B) Find x: log3 x 2
(C) Find b: logb 1,000 3
Solutions
(A) Write y log4 8 in equivalent exponential form:
8 4y
23 22y
Write each number to the same base 2.
2y 3
Recall that b m b n if and only if m n.
y 32
Thus,
3
2
log4 8.
(B) Write log3 x 2 in equivalent exponential form:
x 32
1
1
2
3
9
Thus, log3 ( 19) 2.
(C) Write logb 1,000 3 in equivalent exponential form:
1,000 b 3
103 b 3
b 10
Thus, log10 1,000 3.
Write 1,000 as a third power.
380
5 Exponential and Logarithmic Functions
Matched Problem 3
Find x, b, or y as indicated.
(A) Find y: y log9 27
(B) Find x: log2 x 3
(C) Find b: logb 100 2
• Properties of
Logarithmic
Functions
EXPLORE-DISCUSS 2
The familiar properties of exponential functions imply corresponding properties of
logarithmic functions.
Discuss the connection between the exponential equation and the logarithmic equation, and explain why each equation is valid.
(A) 24 27 211; log2 24 log2 27 log2 211
(B) 213/25 28; log2 213 log2 25 log2 28
(C) (26)9 254; 9 log2 26 log2 254
Several of the powerful and useful properties of logarithmic functions are listed
in Theorem 1.
Theorem 1
Properties of Logarithmic Functions
If b, M, and N are positive real numbers, b 1, and p and x are real numbers,
then:
1. logb 1 0
5. logb MN logb M logb N
2. logb b 1
6. logb
3. logb bx x
7. logb Mp p logb M
4. blog x x, x 0
8. logb M logb N
b
M
logb M logb N
N
if and only if
MN
The first two properties in Theorem 1 follow directly from the definition of a logarithmic function:
logb 1 0
since
b0 1
logb b 1
since
b1 b
The third and fourth properties look more complicated than they are. They follow
directly from the fact that exponential and logarithmic functions are inverses of each
5-3 Logarithmic Functions
381
other. Recall from Section 3-8 that if f is one-to-one, then f 1 is a one-to-one function satisfying
f 1[ f (x)] x
f [ f 1(x)] x
and
Applying these general properties to f (x) b x and f 1(x) logb x, we see that
f 1[ f (x)] x
f [ f 1(x)] x
logb [ f (x)] x
(x)
x
logb x
x
bf
logb b x
x
b
1
Properties 5 to 7 enable us to convert multiplication into addition, division into
subtraction, and power and root problems into multiplication. The proofs of these
properties are based on properties of exponents. A sketch of a proof of the fifth property follows: To bring exponents into the proof, we let
u logb M
v logb N
and
and convert these to the equivalent exponential forms
M bu
N bv
and
Now, see if you can provide the reasons for each of the following steps:
logb MN logb bubv logb buv u v logb M logb N
The other properties are established in a similar manner (see Problems 111 and 112
in Exercise 5-3.)
Finally, the eighth property follows from the fact that logarithmic functions are
one-to-one.
We now illustrate the use of these properties in several examples.
EXAMPLE 4
Using Logarithmic Properties
Simplify using the properties in Theorem 1:
(A) loge 1
(D) log10 0.01
Solutions
2
10
(A) loge 1 0
(C) loge e2x1 2x 1
(E) 10log 7 7
e
(B) log10 10 1
(D) log10 0.01 log10 102 2
(F) elog x x2
2
10
Matched Problem 4
(C) loge e2x1
(F) elog x
(B) log10 10
(E) 10log 7
e
Simplify using the properties in Theorem 1:
(A) log10 105
(D) loge emn
(B) log5 25
(E) 10log 4
10
(C) log10 1
(F) elog (x 1)
4
e
382
5 Exponential and Logarithmic Functions
EXAMPLE 5
Using Logarithmic Properties
Write in terms of simpler logarithmic forms:
(A) log b 3x
(D) logb
Solutions
mn
pq
(B) logb
x
5
(C) logb x7
(E) logb (mn)2/3
(F) logb
x8
y1/5
(A) logb 3x logb 3 logb x
(B) logb
logb MN logb M logb N
x
logb x logb 5
5
logb
(C) logb x7 7 logb x
(D) logb
M
logb M logb N
N
logb Mp p logb M
mn
logb mn logb pq
pq
logb
logb m logb n (logb p logb q)
M
logb M logb N
N
logb MN logb M logb N
logb m logb n logb p logb q
(E) logb (mn)2/3 2
3
logb M p p logb M
logb mn
23(logb m logb n)
logb MN logb M logb N
8
(F) logb
x
logb x8 logb y1/5
y1/5
8 logb x Matched Problem 5
M
logb M logb N
N
logb M p p logb M
logb y
Write in terms of simpler logarithmic forms, as in Example 5.
(A) logb
EXAMPLE 6
1
5
logb
r
uv
(B) logb
m
n
3/5
(C) logb
u1/3
v5
Using Logarithmic Properties
If loge 3 1.10 and loge 7 1.95, find:
(A) loge (73)
Solutions
3
(B) loge 21
(A) loge ( 73) log e 7 loge 3 1.95 1.10 0.85
3
(B) loge 21 loge (21)1/3 1
3
loge (3 7) 13 (loge 3 loge 7)
13 (1.10 1.95) 1.02
5-3 Logarithmic Functions
Matched Problem 6
383
If loge 5 1.609 and loge 8 2.079, find:
(A) loge
510
8
(B) loge 85
4
The following example and problem, though somewhat artificial, will give you
additional practice in using the properties in Theorem 1.
EXAMPLE 7
Using Logarithmic Properties
Find x so that logb x table.
2
3
logb 27 2 logb 2 logb 3 without using a calculator or
Solution
First we use properties from Theorem 1 to express the right side as the logarithm of
a single number.
logb x 23 logb 27 2 logb 2 logb 3
logb 272/3 logb 22 logb 3
logb 9 logb 4 logb 3
94
3
logb 12
logb
x 12
Matched Problem 7
CAUTION
272/3 9; 22 4
Properties 5 and 6 of Theorem 1
Property 8 of Theorem 1
Find x so that logb x 23 logb 8 12 logb 9 logb 6 without using a calculator or table.
We conclude this section by noting three common errors:
1.
logb M
logb M logb N
logb N
logb M logb N logb
M
;
N
logb M
cannot be simplified.
logb N
2. logb (M N) logb M logb N
logb M logb N logb MN;
logb (M N) cannot be simplified.
3. (logb M) p p logb M
p logb M logb Mp
(logb M)p cannot be simplified.
384
5 Exponential and Logarithmic Functions
Answers to Matched Problems
(B) 6 361/2
(C) 19 32
(A) 27 33
1
1
) 2
(A) log4 64 3
(B) log8 2 3
(C) log4 (16
3
1
(A) y 2
(B) x 8
(C) b 10
(A) 5
(B) 2
(C) 0
(D) m n
(E) 4
(F) x4 1
(A) logb r logb u logb v
(B) 53 (logb m logb n)
(C) 13 logb u 5 logb v
(A) 14.01 (to 4 significant digits)
(B) 0.1175 (to 4 significant digits)
x2
1.
2.
3.
4.
5.
6.
7.
EXERCISE
5-3
A
Write Problems 45–58 in terms of simpler logarithmic forms
(see Example 5).
Rewrite Problems 1–8 in equivalent exponential form.
45. logb x6y9
1. log2 64 6
2. log6 216 3
3. log10 100,000 5
4. log10 0.000001 6
5. log9 3 6. log16 2 14
1
2
7. log1/ 4 64 3
8. log1/5 25 2
Rewrite Problems 9–16 in equivalent logarithmic form.
10. 10,000,000 107
9. 0.001 103
11. 4 82/ 3
1/4
13. 81
1
3
2/5
14. 32
15. 5 125
47. logb
u3/2
v5/3
48. logb
v7
u8
49. logb
mn
pq
50. logb
mnp
q
51. logb
1
a4
52. logb
1
a3
53. logb c2 d 2
12. 64 163/2
16. 11 121
3
17. log8 8
18. log5 1
19. log0.5 1
20. log0.2 0.2
21. log10 1,000
22. log2 64
23. loge e
3
24. log5 5
25. log10 0.001
26. log10 0.0001
27. 10log
x2
28. 10log
29. e3loge(x1)
30. e2loge(x1)
10
56. logb
58. logb
57. logb
In Problems 17–30, simplify each expression using
Theorem 1.
x
10
4 4
54. logb c d4
u
vw2
55. logb
1
4
46. logb x2/3y4/3
3
x2y
z3
u3v4
w
4
x
y2z3
3
In Problems 59–68, write each expression in terms of a
single logarithm with a coefficient of 1. Example:
logb u2 logb v logb (u2/v).
59. 2 logb x logb y
60. logb m 21 logb n
61. logb w logb x logb y
62. logb w logb x logb y
63. 3 logb x 2 logb y 41 logb z
64.
B
logb w 3 logb x 5 logb y
65. 5( 21 logb u 2 logb v)
Find x, y, or b, as indicated in Problems 31–44.
31. log2 x 3
32. log3 x 2
33. log5 x 2
34. log4 x 3
35. log3 81 y
36. log4 64 y
37. log16 x 32
38. log8 x 23
39. log1/2 8 y
40. log1/3 81 y
41. logb 32 2.5
42. logb 8 0.5
43. logb 0.0001 2
1
3
44. logb 1,000 3
67.
1
5 (2
logb x 3 logb y)
66. 7(4 logb m 31 logb n)
68.
1
3 (4
logb x 2 logb y)
In Problems 69–76, write each expression in terms of logarithms of first-degree polynomials. Example:
logb
(2x 1)3
3 logb (2x 1) 4 logb (3x 5)
(3x 5)4
69. logb [(x 3)5(2x 7)2]
71. logb
(x 10)7
(1 10x)2
70. logb [(5x 4)3(3x 2)4]
72. logb
(x 3)5
(5 x)3
5-4
73. logb
x2
x 1
74. logb
75. logb (x4 x3 20x2)
98. y log2 (x 3)
76. logb (x5 5x4 14x3)
77. log2 (x 5) 2 log2 3
78. log10 (5 x) 3 log10 2
79. 2 log5 x log5 (x2 6x 2)
100. y log2 x 3
101. (A) For f {(x, y) y (12)x 2x}, graph f, f 1, and
y x on the same coordinate system.
(B) Indicate the domain and range of f and f 1.
(C) What other name can you use for the inverse of f ?
81. loge (x 8) loge x 3 loge 2
82. log7 4x log7 (x 1) 21 log7 4
Find the inverse of each function in Problems 103–106.
83. 2 log3 x log3 2 log3 (4 x)
103. f(x) 53x1 4
84. log4 x log4 (x 2) 21 log4 9
104. g(x) 32x3 2
85. 3 logb 2 12 logb 25 logb 20 logb x
105. g(x) 3 loge (5x 2)
logb 4 32 logb 8 2 logb 2 logb x
If logb 2 0.69, logb 3 1.10, and logb 5 1.61, find the
value of each expression in Problems 87–96.
logb 52
87. logb 30
88. logb 12
89.
90. logb 53
91. logb 27
92. logb 16
93. logb 2
94. logb 3
95. logb 0.9
3
99. y log2 x 2
102. (A) For f {(x, y) y (13) x 3x}, graph f, f 1, and
y x on the same coordinate system.
(B) Indicate the domain and range of f and f 1.
(C) What other name can you use for the inverse of f ?
80. log10 (x2 2x 2) 2 log10 (x 2)
3
2
385
97. y log2 (x 2)
x 1
x3
In Problems 77–86, solve for x without using a calculator
or table.
86.
Common and Natural Logarithms
96. logb 1.5
3
106. f(x) 2 loge (5x 3)
107. Explain why the graph of the reflection of the function
y 3x in the line y x is not the graph of a function.
2
108. Explain why the graph of the reflection of the function
y 2x in the line y x is not the graph of a function.
109. Write loge x loge 100 0.08t in an exponential form
that is free of logarithms.
110. Write loge x loge C kt 0 in an exponential form that
is free of logarithms.
C
Graph Problems 97–100.
111. Prove that logb (M/N) logb M logb N under the
hypotheses of Theorem 1.
Check Problems 97–100 with a graphing utility by graphing
the inverse of each function.
112. Prove that logb M p p logb M under the hypotheses of
Theorem 1.
SECTION
5-4
Common and Natural Logarithms
• Common and Natural Logarithms—Definition and Evaluation
• Applications
John Napier (1550–1617) is credited with the invention of logarithms, which evolved
out of an interest in reducing the computational strain in research in astronomy. This
new computational tool was immediately accepted by the scientific world. Now, with
the availability of inexpensive calculators, logarithms have lost most of their importance as a computational device. However, the logarithmic concept has been greatly
generalized since its conception, and logarithmic functions are used widely in both
theoretical and applied sciences.
Of all possible logarithmic bases, the base e and the base 10 are used almost
exclusively. Before we can use logarithms in certain practical problems, we need to