Chapter 12
Exponential and Logarithmic Functions
12.1 Check Points
1.
( ) 42.2(1.56)
x
fx
2. () 3
x
fx
3
() 3 (, )
11
33 3,
27 27
x
xfx xy
3. 1
() 3
3
x
x
fx
(1)
0
1
1
133 1,3
031 0,1
11
13 1,
33
11
x
4. () 3
x
fx and
1
() 3
x
gx
1
2213
() 3 () 3
11
23 3 3
xx
xfx gx
Chapter 12 Exponential and Logarithmic Functions
5. () 2
x
fx and () 2 3
x
gx
22
() 2 () 2 3
11
22 2 33
xx
xfx gx
The graph of
() 2 3
x
gx
is the graph of
() 2
x
fx
shifted up 3 units.
6. 2017 is 39 years after 1978.
0.0325
0.0325(39)
( ) 1145
(39) 1145 4067
x
fx e
fe
12.1 Concept and Vocabulary Check
1.
x
b; (,) ; (0, )
1.
3.4
2 10.556
2.
2.4
3 13.967
7.
2.3
9.974e
8.
3.4
29.964e
11.
3
fx
x
fx
−2
2
2
11
39
3
−1
1
1
11
33
3
Section 12.1 Exponential Functions
12.
1
3
x
fx
x
fx
−2
21 3
3
11
33 27
3
This function matches graph (b).
13.
31
x
fx
x
fx
−2
2
2
11 8
31 1 1
99
3
14.
3
x
fx
x
fx
−2
2
2
11
39
3
15.
3
x
fx
x
fx
−2
22
339
11
This function matches graph (f).
16.
3
x
fx
x
fx
−2
22
339
−1
11
333
17.
4
x
fx
x
fx
−2
2
2
11
416
4
18.
5
x
fx
x
fx
−2
2
2
11
525
5
19.
3
2
x
gx
x
gx
−2
22
324
239
−1
11
322
233
20.
4
3
x
gx
x
gx
−2
22
439
1
1
44
33
2
2
416
39
21.
1
2
x
hx
x
hx
22
Section 12.1 Exponential Functions
22.
1
3
x
hx
x
hx
−2
22
139
9
311
23.
63
0.6 10 5
xx
x
fx
x
fx
−2
22
3525
539
24.
84
0.8 10 5
xx
x
fx
x
fx
−2
22
4525
5416
25.
1
2 and 2
xx
fx gx
1
() 2 () 2
11
242
1
11
xx
xfx gx
The graph of g is the graph of f shifted 1 unit to the
left.
26.
2
2 and 2
xx
fx gx
Chapter 12 Exponential and Logarithmic Functions
2
() 2 () 2
1
21
4
xx
xfx gx
27.
2
2 and 2
xx
fx gx
2
() 2 () 2
11
2416
11
128
xx
xfx gx
28.
1
2 and 2
xx
fx gx
1
() 2 () 2
12 1
24 2
xx
xfx gx
The graph of g is the graph of f shifted 1 unit to the
right.
29.
2 and 2 1
xx
fx gx
() 2 () 2 1
15
244
xx
xfx gx
Section 12.1 Exponential Functions
30.
2 and 2 2
xx
fx gx
() 2 () 2 2
19
244
xx
xfx gx
31.
2 and 2 2
xx
fx gx
() 2 () 2 2
17
244
13
122
xx
xfx gx
32.
2 and 2 1
xx
fx gx
() 2 () 2 1
13
244
xx
xfx gx
33.
3 and () 3
xx
fx gx
() 3 () 3
11
299
11
133
xx
xfx gx
Chapter 12 Exponential and Logarithmic Functions
34.
3 and 3
xx
fx gx
() 3 () 3
1
29
9
xx
xfx gx
y–axis.
35.
1
2 and 2 1
xx
fx gx
1
() 2 () 2 1
11
242
1
10
2
01 1
xx
xfx gx
36.
1
2 and 2 2
xx
fx gx
1
() 2 () 2 2
13
242
xx
xfx gx
37.
1
3 and 3
3
xx
fx gx
1
() 3 () 3
3
11
2927
11
139
1
xx
xfx gx
Section 12.1 Exponential Functions
38.
3 and 33
xx
fx gx
() 3 () 33
11
293
xx
xfx gx
39. a.
25
0.055
10,000 1 13,116.51
2
A
The balance in the account is $13,116.51 after 5
years of semiannual compounding.
b.
12 5
0.055
10,000 1 13,157.04
12
A
40. a.
210
0.065
5000 1 9479.19
2
A
c.
0.065 10
5000 9577.70Ae
41. Monthly Compounding
12 3
0.07
12,000 1 14,795.11
12
A
Continuous Compounding
42. Quarterly Compounding
44
0.0825
6000 1 8317.84
4
A
Semiannual Compounding
43. Domain:
,
Range:
2,
44. Domain:
,
Range:
3,
45. Domain: {x | x is a real number} or
,
47. Domain:
,
Range:
0,
Chapter 12 Exponential and Logarithmic Functions
49.
() 2 () 2
1
24
4
1
12
2
xx
xfx gx
The point of intersection is
0,1
.
50.
11
() 2 () 2
1
28
2
11 4
xx
xfx gx
The point of intersection is
0, 2
.
51.
2
1
24
1
12
x
xy
2
1
24
1
12
y
yx
52.
3
1
29
1
13
01
13
x
xy
3
1
29
1
13
01
13
y
yx
India’s population in 1974 was 574 million.
b.
27
27 574 1.026 1148f
India’s population in 2001 will be 1148 million.
Section 12.1 Exponential Functions
d.
2055 1974 81, find
54.
80
30
80 1000 0.5 157.49f
Chernobyl will not be safe for human habitation by
2066. There will still be 157.5 kilograms of
cesium-137 in Chernobyl’s atmosphere.
56.
5
510, 000 1 0.03
S
57. a.
() 31
(33) 33 31
64
fx x
f
According to the linear model, 64% of high
school seniors applied to more than three
colleges in 2013.
b.
0.0217
() 32.7
x
gx e
58. a.
() 31
fx x
b.
0.0217
0.0217(30)
() 32.7
(30) 32.7
63
x
gx e
ge
According to the exponential model, about 63%
of high school seniors applied to more than three
0
80 20
e
b.
0.5 1
0.5
180 20
80 20 68.522
fe
e
About 68.5% of information is remembered after
one week.
c.
0.5 4
480 20
fe
Chapter 12 Exponential and Logarithmic Functions
60. a. Note 2005 1626 379t. Then
b.
0.05 379
24 4, 074,662,794
rt
APe e
With continuous compounding, the investment
would be worth $4,074,662,794.
61.
0.122 30
90
30 1270
f
62.
0.122 70
90
70 85.5
1270
f
e
Approximately 85.5% of 70-year-olds have some
coronary heart disease.
63. a.
1.5 0
30,000
0120
30, 000
N
e
b.
1.5 3
4.5
30,000
3120
30,000 24,546
120
N
e
e
64. – 68. Answers will vary
b.
b.
Section 12.1 Exponential Functions
71. does not make sense; Explanations will vary.
Sample explanation: The horizontal asymptote is
0.y
72. makes sense
76. false; Changes to make the statement true will vary.
A sample change is: The graphs do not have the
same graph as they do not coincide.
78. true
79. Graph (a) is
1
3
x
y
.
80.
22
cosh sinhxx
22
2222
22
xx xx
xxxxxxxx
ee ee
44
x
81.
ab
Dab
Da b ab
Da Db ab
23(28)
(3)(4)
2328
(3)(4)
11
(3)(4)
xx
xx
xx
xx
xx
Chapter 12 Exponential and Logarithmic Functions
84. There is no method for solving
2y
x
for y.
86.
25
25
fx x
yx
Interchange x and y and solve for y.
12.2 Check Points
1. a.
7
3
3log
7
x
x
2. a.
5
2
2
5log
x
x
c.
36
1
log 6 2
because
1
2
36 36 6.
5. a. Because
log ,
x
b
bx
we conclude
8
7
log 7 8.
11
93
210123
() 3 1 3 9 27
x
x
fx
Reverse these coordinates to obtain the coordinates
of
() log .gx x
7.
50
x
29 48.8log11
80
A 10-year-old boy has attained approximately 80%
of his adult height.
40.x
40
4
x
x
Section 12.2 Logarithmic Functions
b. The domain of g consists of all x for which
2
12.2 Concept and Vocabulary Check
1.
y
bx
2. logarithmic; b
3. 1
4. 0
12.2 Exercise Set
1.
2
4
4log16
216
5.
5log32
b
5125
9.
3
2
28
log 8 3
10.
4
5
5625
log 625 4
11.
4
1
216
14.
3
1
3
64
64 4
64 4
1
log 4 3
Chapter 12 Exponential and Logarithmic Functions
18.
3
343
log 343 3
b
b
21.
4
2
log 16
416
44
2
y
y
y
y
24.
3
3
log 27
327
33
3
y
y
y
y
27.
2
1
log 8
1
y
y
28.
3
1
log 9
1
39
1
33
y
y
y
30.
6
1
2
log 6
66
66
1
y
y
y
Section 12.2 Logarithmic Functions
32.
3
1
log 3
y
33.
64
1
2
log 8
64 8
64 64
1
2
y
y
y
y
36.
11
1
log 11
11 11
1
y
y
y
37.
4
0
log 1
41
44
0
y
y
y
y
40.
6
4
6
log 4
44
y
y
4
log
gx x
45.
1
2
1
2
log
x
fx
gx x
Chapter 12 Exponential and Logarithmic Functions
46.
1
4
x
fx
47.
5
log 4fx x
40
4
x
x
The domain of f is
4, .
49.
5
log 2fx x
20
2
2
x
x
x
The domain of f is
,2 .
50.
5
log 7fx x
52.
2
ln 7fx x
54.
3
log1000
10 1000
10 10
3
y
y
y
y
7
8
y
57. Since
log
10 ,
x
x
log33
10 33.
58. Since
log
10 ,
x
x
we conclude that
log53
10 53.
59.
0
ln1
1
0
y
y
y
e
ee
y
Section 12.2 Logarithmic Functions
63.
6
6
1
ln ln
e
e
65. Since
ln
,
x
ex
ln125
125.
e
66. Because
ln
,
x
ex
we conclude that
ln 300
300.
e
67. Since
ln ,
x
ex
9
ln 9 .
x
ex
73.
3
2
log 1 2
31
91
10
x
x
x
x
The solution set is {10}.
76.
64
2
log 3
x
78.
5
52 52
5
log log 32 log log 2
log 5 1
79.
4
23 23
log log 81 log log 3
shifted up 2 units.
84. (a) The graph is similar to that of lnyx, but
shifted down 2 units.
85. (b) The graph is similar to that of lnyx, but
reflected across the y-axis and then shifted right 1
unit.
Chapter 12 Exponential and Logarithmic Functions
89. a. 2008 is 29 years after 1979.
6.3ln 12.8
fx x
90. a. 2000 is 21 years after 1979.
6.3ln 12.8
21 6.3ln 21 12.8
32
fx x
f
91.
12 6
18
10log 10 6.3 10
10log 6.3 10 188.0
D
The decibel level of a blue whale is approximately
188 decibels. At close range, the sound could
rupture the human ear drum.
93. a. The original exam was at time, t = 0.
b.
28815ln21
88 15ln 3 71.5
f
88 15ln 11 52.0
12 88 15ln 12 1
88 15ln 13 49.5
f
c.