Chapter 11
FUNCTIONS
18.
()
8
hx ⎛⎞
=⎟
⎜⎟
⎜
⎝⎠
a.
()
22
18
264
81
h
−
⎛⎞ ⎛⎞
⎟⎟
⎜⎜
−= = =
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝⎠
(
)
a.
()
41 3
3
11
44 4 64
4
g−+ −
−= = = =
(
)
(
)
28.
(
)
fx e+
=
(
)
2 148.413fe e
==≈
(
)
() ( )
2
4,
11
24 2,
16 16
x
xfx xy
−
=
⎛⎞
⎟
⎜
−=−
⎟
⎜⎟
⎜
⎝⎠
222
32.
()
3
fx ⎛⎞
⎟
⎜
=⎟
⎜⎟
⎜
⎝⎠
() ()
3
2
1,
3
1
11 1
11,
33 3
11 1
22,
39 9
x
xfx xy
−
⎛⎞
⎟
⎜
=⎟
⎜⎟
⎜
⎝⎠
⎛⎞
⎛⎞ ⎛ ⎞
⎟⎟
⎜⎜
=
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝ ⎠
⎛⎞ ⎛ ⎞
⎟⎟
⎜⎜
=
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝ ⎠
34.
(
)
3x
fx=−
() ( )
()
()
3
3
2
3
3,
11 1
33 3,
27 27
3
2392,9
3 3 27 3, 27
x
xfx xy
−
=−
⎛⎞
⎟
⎜
−− =−=− −−⎟
⎜⎟
⎜
⎝⎠
−=− −
−=− −
36.
(
)
3x
fx +
=
() ( )
1
31 2
2
3,
11 1
33 3 3,
99
3
x
xfx xy
+
−+ −
=
⎛⎞
⎟
⎜
−===−
⎟
⎜⎟
⎜
⎝⎠
Section 11.1 Exponential Functions
() ()
1
13,
64 3 61
1
444
11,
4
11
4
x
xfx xy
−
⎛⎞
⎟
⎜
=−
⎟
⎜⎟
⎜
=−=
⎛⎞
⎜
⎝⎠ ⎟
⎜−⎟
⎜⎟
⎜
⎝⎠
=−
64
⎟
⎜−= −
=−
40. 4
10 10,000 10 4
xx==⇒=
44. 3
125 5 5 3 3
== ⇒=−⇒=−
()
25
33
x
=
48. 3
35
232
x
x
−
−
=
50. 2
61
x
+
=
52.
()
()
()
31 4
33
314
x
x
−
=
−=
54.
()
()
3
52 3
32 8
22
x
−+ −
=
=
() ()
()
()
51
8
22
85 1
x
x
xx
+
=
=+
224
58.
() ()
125 625
=
1
1
1
2
11 1
x
−
⎛⎞
⎟
⎜⎟
⎜
⎝⎠
⎛⎞ ⎛ ⎞
62. Exponential function: it has a variable in
the exponent.
64.
(
)
()
51
151 1
fx
f−
=− +
⎟
⎜
−=− +=− +=−+=
⎟
⎜⎟
66.
(
)
(
)
(
)
512500.85 555
v
=≈
0 1250 0.85 1250 1 1250v===
(
)
(
)
821.1 4.3
h
=≈
72.
()
30 2
At ⎛⎞
⎜⎟
⎜
⎝⎠
130 65
1
(
)
76.
(
)
(
)
500 3 t
mt=
(
)
() ()
5
53
500 3 13,500
33
t
t
=
=
225
1. a.
23
224 39
x
==
xx
xx
2. The restriction 0b> guarantees that the
3. Domain of
(
)
2. The term logbx may be read as “the logarithm of
4. The graph of a logarithmic function is symmetric
8. 2
4
log 16 2 4 16=⇒ =
−
16. 13
64
18. 15
30. 12
15
226
32.
(
)
22
log 64 log 2 6==
(
)
110 110
48. 3
164 164
11
log log
464
=
5
3
1
x
x
−
=
=
x
60.
log 81 2
x
=
()
81
x
x
=
=
()
14
1
16
x
=
227
()
21
33
x
=
70.
()
16
23
44
x
=
=
72.
()
12
5
22
55
x
xx
−
=
−= ⇒ =−
74.
(
)
4
log 4y
fx y x x== ⇒=
()
()
2
2
4,
111
2 4 16 16, 2
y
yx xy
−
=
⎛⎞
=
()
()
0
2
4
4
1
011,0
1111
yx xy
⎟
⎜
=⎟
⎜⎟
⎜
⎝⎠
⎜⎟
⎜
⎝⎠
⎛⎞
⎟
⎜=
⎟
⎜⎟
⎛⎞ ⎛ ⎞
64 64
78.
(
)
2
log 2 y
fx y x x −
==− ⇒=
()
()
()
()
()
()
()
()
33
22
11
0
2,
32 2 8 8,3
22 2 4 4,2
12 2 2 2,1
021 1,0
y
yx xy
−
−−
−−
−−
−
=
−==−
−==−
−==−
=
228
2
2
25
64
25
x
=
x=
82.
(
)
4
log 4 y
fx y x x −
==− ⇒=
()
()
()
22
4,
24 4 16 16,2
y
yx xy
−
−−
=
−==−
90. 10
0
⎜⎟
⎜
⎝⎠
()
()
()
10
6
10
20log 10
20 6
=
=
92. 2
⎜⎟
⎜
⎝⎠
2
1
1600log 8
1
⎛⎞
⎟
⎜
=− ⎟
⎜⎟
⎜
⎝⎠
⎛⎞
⎟
⎜
94. 10
10logtN=−
10
1
10log 10
t
⎛⎞
⎟
⎜
=− ⎟
⎜⎟
⎜
⎝⎠
229
1. The value of y can never be 0 because
(
)
1000
−
40. 3
54. 7
Since , 7 20.
bx==
58. log log 23
Since , 23.
bxn==
10 10
log 27 log 2
=+
62. 23
3333
33
log 64 log 125
=−
log log
b
=+
()
log 4 log 6 log 2
log 64 log 36 log 16
log 64 log 36 log 16
xxx
xxx
xxx
=−−
=−−
=− +
68.
()
77
16log log
u
70.
(
)
()
(
)
22
2
54
2
4log log 3
log 3
nn
nn
++
=+
72.
(
)
()
22
2ab ab
−− −
⎢
⎥
⎣
⎦
1log
⎢
⎥
⎣
⎦
74. 888
4 log 5 log 2 log
bac
−+−
76. 777
33
+−
[]
()
7
77 7
77
12 log l og log
3
uv w
=+−
82. 33
3333
log 3 log 3 log log
uv u v
=++
2
55
3
1log 9 2 log
v
⎡
⎤
=+
⎢
⎥
⎣
⎦
231
86.
555
4
log log log
b
=−
2
22
33
1log
3
x
xy
⎡⎤
⎡⎤
=−
⎢⎥
⎣⎦
zz
(
)
()
14
32
32
1log log 5 log
log log 5log
44
nn
nnn
ac
acb
=+−
55 55
44
100. 4
102.
55 5
22 2
log log log
4
1log log
yy y
==
⎜⎟
⎜⎢⎥
⎟
⎟
⎜
⎝⎠ ⎣⎦
104. 555
25
()
(
)
10
2.5 8 20
=− − =
()
10 10
I
I
⎢
⎥
⎣
⎦
110. a.
(
)
10 10 10
log log log 1.05
t
APt
=+
b.
(
)
(
)
1000 1.05 1000 1.05 1050A===
232
is log .
byx=
So log .
x
bbx=
Alternate proof:
2. 33
3.3
=−
3.2
=
and Change of Base
Exercises
2. The inverse of the natural exponential
log 0.000001 log10 6
==−
e
1.2691
≈
1.2619
≈−
66.
(
)
80 8ln 1sx=− +
c
35 7
b
c
234
(
)
(
)
()
5
2log 3
2
53
69 3
x
x
xx x
+
=+
++=+
()
()
5
?
2log 2 3
?
523
−+ =− +
5
?
2log 0
50
=
Logarithmic Equations
2. 656
log 6 log 56
x
=
=
4. 115
3
1
2.4650
x
x
⎟
⎜⎟
⎜
⎝⎠
⎛⎞
6. 6
77
x
=
Section 11.5 Exponential and Logarithmic Equations 235
10. 2
57.5
=
log1000 log 95
3 log 95
x
x
−=
−⋅ =
14.
1
268
log 2 log 68
x
+
=
=
16.
35
6324
log 6 log 324
x
−
=
=
log 6
x
(
)
2
24.
(
)
3
2
13
x
−
−=
(
)
3
24
2
6
log 12 6
x
=
30. 55
log log 2 1
x
−=
236
32.
(
)
()
33
3
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
−+ + − =
−+ −=
Check 3.x=
?
()()
7
log 3 3 1
xx
⎡⎤
−+=
⎣⎦
()()
77
log 4 3 log 4 3 1
−− + −+ =
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
++ −=
⎡⎤
+−=
⎣⎦
15
x
=−
23
Check .
15
x=−
log 4 log 9 1
53
⎟⎟
⎜⎜
−++ −−=
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠⎝⎠
()
()
?
38.
(
)
22
10 4 16
10 4 16
x
x
x
xx
+=
+=
237
(
)
(
)
55
5
1
log 2 1 log 8 1
21
log 1
8
21
xx
x
x
x
−− − =
−=
−
−=
()
()
9
32
log 6 2
69
xx
xx
⎡⎤
−=
⎣⎦
−=
Check 3.x=−
?
3
Check 9.x=
The solution is 9.
44.
(
)
(
)
33
log 9 log 2 3 2xx−− + =−
3
9
log 2
23
93
x
x
x
−=−
+
−=
46. 44
4
log log 3 2
log 2
3
x
x
=
()
(
)
(
)
44
240
xx
+−=
Check 2.x=−
Check 4.x=
()
?
44
?
44
3
log 4 2 log 4 2
3
log 2 log 4 2
−+ =
+=
238
(
)
25
log 4 log 75
2 5 log 4 log75
x
x
+
=
+=
9
t
≈
54. 0
VVe
=
31,000
2ln
31
k
e
k
=
=
The car will be worth approximately $11,000.
56.
0
⎜⎟
⎜
⎝⎠
12
140 10log 10
I
−
⎛⎞
⎟
⎜
=⎟
⎜⎟
⎜
⎝⎠
N when 0.y=
log 3 1.6
y
=≈
62.
()
0
k
=−
k is 0.231;− it represents the rate at which
239
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
intersection of the graphs. The x-coordinate
of any point of intersection is the solution.
The solution is approximately 0.8617.
3log2
ln 2 ln
ln 2
e
t
r
=
=
0.6931 69 72 72 9 yr.
[
)
2 2,
x
≥∞
b.
(
)
(
)
2
1
log 1 1
12
3 3,
x
x
x
−>
−>
>∞
()
()
Let 4. Let 0.
=− =
()
?
Let 4.
log17 1
x
=
≤
