Chapter 11
EXPONENTIAL AND LOGARITHMIC
FUNCTIONS
11.1 Exponential Functions
Exercises
2. The number e is irrational.
)
10. Exponential function: it has a variable in
the exponent.
12. Rational function: it has a variable in the
denominator.
)
)
18.
()
1
8
x
hx ⎛⎞
⎟
⎜
=⎟
⎜⎟
⎜
⎝⎠
20.
(
)
1
4x
gx +
=
a.
()
41 3
3
11
44 4 64
4
g−+ −
−= = = =
b.
(
)
11 2
14 416g+
===
c.
(
)
31 4
3 4 4 256g+
===
24.
()
fx e
=
a.
(
)
()
33
3 20.086fee
−−
−= = ≈
b.
(
)
1
10.368fe
−
=≈
4x
2
⎝⎠
30.
(
)
4x
fx=
Chapter 11 Exponential and Logarithmic Functions
222
30. (continued)
32.
()
1
3
x
fx ⎛⎞
⎟
⎜
=⎟
⎜⎟
⎜
⎝⎠
() ()
()
3
3
1,
3
1
3 3 27 3,27
3
x
xfx xy
−
⎛⎞
⎟
⎜
=⎟
⎜⎟
⎜
⎝⎠
⎛⎞
⎟
⎜
−==−
⎟
⎜⎟
⎜
⎝⎠
34.
(
)
3x
fx=−
() ( )
()
()
()
()
3
0
1
2
3
3,
11 1
0310,1
1331,3
2392,9
3 3 27 3, 27
x
xfx xy
−
=−
⎛⎞
⎟
⎜
−=− −
−=− −
−=− −
−=− −
()
()
()
01 1
11 2
21 3
0333 0,3
1339 1,9
23327 2,27
+
+
+
==
==
==
Section 11.1 Exponential Functions
223
38.
()
13
4
x
fx ⎛⎞
⎟
⎜
=−
⎟
⎜⎟
⎜
⎝⎠
() ()
1
13,
4
1112
311
444
11,
4
11
4
x
xfx xy
⎛⎞
⎟
⎜
=−
⎟
⎜⎟
⎜
⎝⎠
⎛⎞
⎟
⎜−= −
⎟
⎜⎟⎛⎞
⎜
⎝⎠ ⎟
⎜−⎟
⎜⎟
⎜
⎝⎠
=−
44. 3
125 5 5 3 3
== ⇒=−⇒=−
46.
()
25
25
9243
33
33
5
x
x
x
=
=
=
()
()
()
1
34
31 4
33
33
314
x
x
x
−
−
=
=
−=
()
22
523
510 3
13
513 5
x
x
xx
=
−+ =−
−+=−
−=−⇒=
56.
() ()
()
1
1
85
51
8
256 32
22
22
xx
xx
x
x
+
+
+
=
=
=
Chapter 11 Exponential and Logarithmic Functions
224
58.
11
125 625
xx
+−
=
60.
()
1
2
x
fx ⎛⎞
⎟
⎜
=− ⎟
⎜⎟
⎜
⎝⎠
() ()
()
3
3
1,
2
1
3283,8
x
xfx xy
−
⎛⎞
⎟
⎜
=− ⎟
⎜⎟
⎜
⎝⎠
⎛⎞
⎟
⎜
−− =−=− −−
⎟
⎜⎟
⎜
⎝⎠
62. Exponential function: it has a variable in
the exponent.
64.
(
)
51
x
fx
=− +
68. a.
(
)
(
)
(
)
(
)
5
1250 0.85
512500.85 555
t
vt
v
=
=≈
After 5 yr, the value will be about $555.
b.
(
)
(
)
(
)
0
0 1250 0.85 1250 1 1250v===
The value was $1250.
After 130 days, 7.5 mg will remain.
74.
(
)
0.3
300 t
Pt e=
)
)
()
0.3 4 1.2
()
() ()
5
53
327
33
t
t
=
=
Section 11.2 Logarithmic Functions
225
Mindstretchers
1. a.
44
44
23
11 11
42 3
16 81
xx
x
−−
−== ==
b. When x is negative, the value of 2x
is greater than the value of 3 .
x
When x is positive, the value of 2x
is less than the value of 3 .
x
c. i.
(
)
23; ,0
xx
>−∞
ii.
(
)
23;0,
xx
<∞
2. The restriction 0b> guarantees that the
value of x
b is a real number for all values
of .x The restriction 1b≠ guarantees that
all exponential functions are one-to-one
functions.
11.2 Logarithmic Functions
Exercises
12. 4
10
11
log 4 10
10,000 10,000
−
=− ⇒ =
14.
2
16
1
log 36 2 36
6
−
⎛⎞
⎟
⎜
=− ⇒ =
⎟
⎜⎟
⎜
⎝⎠
1
22.
2
15
11 1
log 2
525 25
⎛⎞
⎟
⎜=⇒ =
⎟
⎜⎟
⎜
⎝⎠
24. 2
9
11
9log2
−=⇒ =−
Chapter 11 Exponential and Logarithmic Functions
226
32.
(
)
6
22
log 64 log 2 6==
)
40.
()
1
22
1
log log 2 1
2
−
==−
42.
(
)
3
110 110
log 1000 log 10
=
50. 88 8
13
3
13
8
11 1
log log log
28
8
1
log 8 3
−
==
==−
52. 2
6
log 6
2
x
x
=
=
56. 16
2
log 2
1
x
−
=−
⎛⎞
13
3
64
64
4
x
x
x
=
=
=
60.
log 81 2
x
=
()
3
13
13
3
3
27
8
27
8
27 3
82
x
x
x
=
⎛⎞
⎟
⎜
=⎟
⎜⎟
⎜
⎝⎠
==
Section 11.2 Logarithmic Functions
227
68.
9
log 3
x
=
70.
16
log 64
16 64
x
x
=
=
72.
12
log 32
x
=
74.
(
)
4
log 4y
fx y x x== ⇒=
()
2
2
1
4,
111
24 ,2
16 16
4
11
14 ,1
44
y
yx xy
−
−
=
⎛⎞
⎟
⎜
−== −
⎟
⎜⎟
⎜
⎝⎠
⎛⎞
⎟
⎜
−= −
⎟
⎜⎟
⎜
⎝⎠
y
⎛⎞
()
2
2
1
2
1
241616,2
4
1,1
44 4
1111
−
−
⎛⎞
⎟
⎜
−==−
⎟
⎜⎟
⎜
⎝⎠
⎜⎜
=
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝ ⎠
⎛⎞ ⎛ ⎞
Chapter 11 Exponential and Logarithmic Functions
228
78. (continued)
80.
25
log 2
64
x
=
8
82.
(
)
4
log 4 y
fx y x x −
==− ⇒=
()
()
()
22
4,
24 4 16 16,2
y
yx xy
−
−−
=
−==−
84.
()
14
4
81 81 81
11
log 3 log 81 log 81
44
== = =
86. 1/4
81
1
log 3 81 3
=⇒ =
10 5
5
20
20log 210
20log 10 10
S−
⎛⎞
⎟
⎜
=⎟
⎜⎟
⎜
⎝⎠
×
=×
2
2
5
1600log 40
1
1600log 8
1
1600log 2
t
⎛⎞
⎟
⎜
=− ⎟
⎜⎟
⎜
⎝⎠
⎛⎞
⎟
⎜
=− ⎟
⎜⎟
⎜
⎝⎠
⎛⎞
⎟
⎜
=− ⎟
()
10
1
10
1
10log 10
10log 10
10 1
t
−
⎛⎞
⎟
⎜
=− ⎟
⎜⎟
⎜
⎝⎠
=−
=− −
Section 11.3 Properties of Logarithms
229
Mindstretchers
1. The value of y can never be 0 because
there is no real number x such that 0,
x
b=
where 0.b≠
11.3 Properties of Logarithms
Exercises
2. The logarithm of a quotient is the logarithm
of the numerator minus the logarithm of the
denominator.
4. For any base, the logarithm of 1 is 0.
10. 222
2
22
2
4
log log 4 log 9
9
log 2 log 9
2log9
=−
=−
=−
12. 555
log 15 log 15 logyy=+
20. 444
log log log
qqp
p=−
22.
() ()
1
log log 1 log 5
5
kk k
kkk
k
+=+−+
+
30. 7
log 7 1 log 7 log log
aaaa
aa
−= − =
32.
(
)
(
)
(
)
(
)
333
3
log 2 log 6 log 2 6
log 6 12
yy
y
++ = +
=+
34. log log log log
bb b b
ca ca ac+= =
36. 66 6
log log log y
yx x
−=
44. 14
41
log log log
4
xx x
yy y==
46. 2
77
log 2logyy
−=−
48. 7
3
Since log , log 3 7.
x
bbx==
Chapter 11 Exponential and Logarithmic Functions
230
60.
()
3
10 10 10 10
10 10
10
10
3log 3 log 2 log 3 log 2
log 27 log 2
log 27 2
log 54
+= +
=+
=⋅
=
⎢
⎢
⎢
⎣
66.
()
324
3log 4 2 log 6 4 log 2
log 4 log 6 log 2
log 64 log 36 log 16
log 64 log 36 log 16
64
log 36 16
1
log 9
xxx
xxx
xxx
xxx
x
x
−−
=−−
=−−
=− +
=⋅
=
70.
(
)
()
()
()
(
)
22
4
22
4
2
54
2
4log log 3
log log 3
log 3
log 3
nn
nn
nn
nn
++
=++
=+
=+
72.
(
)
()
22
22
1log log
2ab ab
⎡
⎤
−− −
⎢
⎥
⎣
⎦
22
2
1log
2
ab
ab
⎡
⎤
−
⎢⎥
=⎢⎥
−
⎣⎦
⎡
⎤
76. 777
21
log log 7 log
33
uvw
+−
[]
7
77 7
27
777
27
77
1
12 log log log
3
1log log log
3
1log log
3
uv w
uvw
uv w
=+−
⎡⎤
=+−
⎢⎥
⎣⎦
⎡⎤
=−
⎢⎥
⎣⎦
84.
(
)
[]
13
322
55
2
5
2
55
55
55
log 9 log 9
1log 9
3
1log 9 log
3
1log 9 2log
3
12
log 9 log
33
vv
v
v
v
v
=
=
⎡
⎤
=+
⎢
⎥
⎣
⎦
=+
=+
Section 11.3 Properties of Logarithms
231
86.
3
34
555
4
55
log log log
3log 4log
aab
b
ab
=−
=−
90.
54
542
2
log log log log
5log 4log 2
zzzz
zz
xy xyz
z
xy
=+−
=+−
94.
432
5
logn
ac
b
(
)
()
14
32
5
14
32 5
32
log
log log
1log 5log
4
n
nn
nn
ac
b
ac b
ac b
=
=−
=−
98.
()
55 55
14 2
55
11
log 8log log 2 log
44
log log
ab ab
ab
−
−+ =− +
=+
55
log log
42
xy
=−
104. 555
log log log 25
25
bb
=−
106.
()
()
(
)
10
8
10
8
10
2.5log
2.5log 1.0 10
2.5log 10
2.5 8 20
mb
m−
−
=−
=− ×
=−
=− − =
The apparent magnitude of Vega is 20.
108.
10 16
16
10log 10
I
L
−
−
=
⎡
⎤
Chapter 11 Exponential and Logarithmic Functions
232
Mindstretchers
1. a. Let .
x
yb=
Then, the equivalent logarithmic equation
is log .
byx=
So log .
x
2. 33
2
33
33
3
3
3 log 2 1 2 log 2
1 log 3 log 2
1 log 9 log 2
1log92
1log18
+=++
=+ +
=+ +
=+ ⋅
=+
3. log 3.5, log 0.2, log 1.4
bb b
xyz===
a.
(
)
log log log
3.5 0.2
3.7
bbb
xy x y=+
=+
=
3.2
=
11.4 Common Logarithms,
Natural Logarithms,
and Change of Base
Exercises
8.
log 22 1.3424≈
10. 1
log 0.8451
7≈−
12. log 0.9 0.0458≈−
14. ln 6 1.7918≈
16. ln13 2.5649≈
18. 3
ln 0.2877
4
⎛⎞
⎟
⎜≈−
⎟
⎜⎟
⎜
⎝⎠
Section 11.4 Common Logarithms, Natural Logarithms, and Change of Base
233
36.
2
3232
ln ln 3
ee==
38. ln y
ey=
48. 13
ln 4 log 4
log 4 11
ln log
33
1.2619
==
≈−
56. 6
log10 6=
58. ln 7 7e=
64.
0
15.2ln P
tP
⎛⎞
⎟
⎜⎟
=⎜⎟
⎜⎟
⎜
⎝⎠
50,000
15.2ln 11
24,342
t⎛⎞
⎟
⎜
=≈
⎟
⎜⎟
⎟
⎜
⎝⎠
00
ln ln
c
WW
cWW
⎛⎞
⎟
⎜⎟
==
⎜⎟
⎜⎟
⎜
⎝⎠
70. 22
51
5730log 5730log
35 7
−=−
log
10 50 ln 50 3.9120
3.9120
ln 50
10 50 log50 1.6990 ln10
e
e
e
≈
≈
≈
Chapter 11 Exponential and Logarithmic Functions
234
2.
(
)
3fx x=+
()
(
)
()
)
)
5
5
2log
2log 3
5
53
fx
x
fx
x
+
=
=+
Check 2.
x=−
()
()
5
?
2log 2 3
523
−+ =− +
Check 3.
x=−
()
()
5
?
2log 3 3
533
−+ =− +
3. The change-of-base formula can be used to
rewrite the expression on the right side of the
equation in terms of common logarithms or
natural logarithms. So the equation becomes:
11.5 Exponential and
Logarithmic Equations
4. 115
3
2.4650
x
x
⎛⎞
⎟
⎜=
⎟
⎜⎟
⎜
⎝⎠
≈−
log 7 log 7
6
log 7
log 7
x
x
x
=
=
Section 11.5 Exponential and Logarithmic Equations 235
10. 2
2
57.5
log5 log 7.5
x
x
=
=
12. 1000 95
log1000 log 95
x
x
−
−
=
=
14.
1
1
268
log 2 log 68
x
x
+
+
=
=
16.
(
)
35
35
6324
log 6 log 324
3 5 log 6 log 324
x
x
x
−
−
=
=
−=
18. 0.012
0.012
7.5
ln ln 7.5
x
x
e
e
−
−
=
=
20.
(
)
2
5
log 7 5
72
x
x
+=
+=
369
23
x
x
=
=
26.
(
)
2
3
24
log 19 4
19 3
x
x
−=
−=
28. 22
2
6
log log 12 6
log 12 6
12 2
12 64
x
x
x
x
+=
=
=
=
2
5
2
x
=
Chapter 11 Exponential and Logarithmic Functions
236
32.
(
)
()
33
log 6 log 3
log 6 3
xx
xx
++ =
⎡⎤
+=
Check 9.x=−
()()
(
)
(
)
?
33
?
33
log 9 6 log 9 3
log 3 log 9 3
−+ + − =
−+ −=
34.
(
)
(
)
()()
77
7
1
log 3 log 3 1
log 3 3 1
337
xx
xx
xx
−+ +=
⎡⎤
−+=
⎣⎦
−+=
Check 4.x=
() ()
?
77
log 4 3 log 4 3 1
−+ +=
36.
(
)
(
)
()()
log 3 4 log 5 9 1
log 3 4 5 9 1
xx
xx
++ −=
⎡⎤
+−=
23
15
x
=−
23
Check .
15
x=−
Check 2.x=
()
()
()
()
() ( )
?
?
log 3 2 4 log 5 2 9 1
log 6 4 log 10 9 1
++ −=
++ −=
10 4 16
10 4 16
x
x
x
xx
+=
+=
Section 11.5 Exponential and Logarithmic Equations
237
40.
(
)
(
)
55
log 2 1 log 8 1
xx
−− − =
42.
()
()
99
9
3
log 6 log 2
3
log 6 2
xx
xx
−+ =
⎡⎤
−=
⎣⎦
() ()
?
99
3
log 9 log 3 2
−+ −=
Logarithms of negative numbers are undefined.
3− is an extraneous solution.
Check 9.x=
44.
(
)
(
)
33
log 9 log 2 3 2xx−− +=−
784
12
x
x
=
=
46. 44
log log 3 2
x
x
−=
()
()
(
)
(
)
4
3/2
2
2
24
28
280
240
xx
xx
xx
xx
⎣⎦
−=
−=
−−=
+−=
()
?
44
?
44
?
3
log 4 2 log 4 2
3
log 2 log 4 2
13
1
22
33
22
−+ =
+=
+=
=
The solution is 4.
Chapter 11 Exponential and Logarithmic Functions
238
50.
25
25
475
log 4 log 75
x
x
+
+
=
=
52.
(
)
1334 1.073 t
E=
(
)
()
1334 1.073 2500
2500
1.073 1334
t
t
=
=
54. 0
0
16,000, 31,000, 2
kt
VVe
VV t
=
===
()
2
2
31, 000 16,000
16,000
31, 000
k
k
e
e
=
=
56.
0
10log I
LI
⎛⎞
⎟
⎜⎟
=⎜⎟
⎜⎟
⎜
⎝⎠
58. a.
(
)
1
1
ln 1
r
r
Re
Re
rR
=−
+=
=+
21
27
log 2 log 3
log 2 log 3
log 3 1.6
log 2
y
y
y
y
=
=
=
=≈
Section 11.5 Exponential and Logarithmic Equations
239
Mindstretchers
1. a. Enter the expression 3
2x into Y1 and 6
in Y2, and then, graph. Use the TRACE or
INTERSECT feature to find the point(s) of
b. 3
3
26
log 2 log 6
3log2 log6
x
x
x
=
=
=
2.
2
2
ln 2 ln
rt
rt
rt
rt
APe
PPe
e
e
=
=
=
=
3. a.
[
)
3
39
log 9
2 2,
x
x
x
≥
≥
≥∞
b.
(
)
log 1 1
x
−>
2
2
90
9
3
x
x
x
−=
=
=±
01
1.2304 1
True
False
≤
≤
?
Check 3.
Let 4.
x
x
>
=