1028 3 342 2 f 342 f 7 3 1 f g 7 f g 7 108 therefore

1028
3 342 2
f 342
f 7 3 1
f g 7 f g 7 108.
Therefore,
7 3 1 344.
g f 3 3 3 2 3 1
g f ( x g f x g 3 x 2 3 x 2 3 1
(b) Increasing on
Decreasing on
(c) Relative minimum:
0, 0
, 0
0,

Section 3.3 Properties of Logarithms 281
12. log
1 4
5
log 5
log 1 4
ln 5
ln 1 4
1.161 13. log
9
0.4
log 0.4
log 9
ln 0.4
ln 9
0.417
14. log
20
0.125
log 0.125
log 20
ln 0.125
ln 20
0.694 15. log
15
1250
log 1250
log 15
ln 1250
ln 15
2.633
16. log
3
0.015
log 0.015
log 3
ln 0.015
ln 3
3.823 17. log
4
8
log
2
8
log
2
4
log
2
2 3
log
2
2
2
3
2
Section 3.3 Properties of Logarithms
■ You should know the following properties of logarithms.
(a)
(b)
(c)
(d)
■ You should be able to rewrite logarithmic expressions using these properties.
ln u n n ln u log
a
u n n log
a
u
ln
u
v
ln u ln v log
a
u
v
log
a
u log
a
v
ln uv ln u ln v log
a
uv log
a
u log
a
v
log
a
x
ln x
ln a
log
a
x
log
10
x
log
10
a
log
a
x
log
b
x
log
b
a
Vocabulary Check
1. change-of-base 2.
3. 4.
This is the Product Property. Matches (c). This is the Power Property. Matches (a).
5.
This is the Quotient Property. Matches (b).
log
a
u
v
log
a
u log
a
v
ln u n n ln u log
a
uv log
a
u log
a
v
log x
log a
ln x
ln a
1. (a)
(b) log
5
x
ln x
ln 5
log
5
x
log x
log 5
2. (a)
(b) log
3
x
ln x
ln 3
log
3
x
log x
log 3
3. (a)
(b) log
1 5
x
ln x
ln 1 5
log
1 5
x
log x
log 1 5
4. (a)
(b) log
1 3
x
ln x
ln 1 3
log
1 3
x
log x
log 1 3
5. (a)
(b) log
x
3
10
ln 3 10
ln x
log
x
3
10
log 3 10
log x
6. (a)
(b) log
x
3
4
ln 3 4
ln x
log
x
3
4
log 3 4
log x
7. (a)
(b) log
2.6
x
ln x
ln 2.6
log
2.6
x
log x
log 2.6
8. (a)
(b) log
7.1
x
ln x
ln 7.1
log
7.1
x
log x
log 7.1
9. log
3
7
log 7
log 3
ln 7
ln 3
1.771
10. log
7
4
log 4
log 7
ln 4
ln 7
0.712 11. log
1 2
4
log 4
log 1 2
ln 4
ln 1 2
2.000

282 Chapter 3 Exponential and Logarithmic Functions
18.
4 4 log
2
3
4 log
2
2 4 log
2
3
2 log
2
2 2 4 log
2
3
2 log
2
4 4 log
2
3
log
2
4
2
3
4
log
2
4
2
log
2
3
4
19.
3 log
5
2
log
5
5 3 log
5
2 1
log
5
1
125
log
5
1
2
log
5
1
250
log
5
1
125
1
2
20.
log 3 2
log 3 2 log 10
log 3 log 10 2
log 3 log 100
log
9
300
log
3
100
21.
6 ln 5
ln 5 6
ln 5 e
6
ln 5 ln e
6 22.
ln 6 2
ln 6 2 ln e
ln
6
e 2
ln 6 ln e 2
23. log
3
9 2 log
3
3 2
24. log
5
1
125
log
5
5 3 3 log
5
5 3 1 3
25. log
2
4
8
1
4
log
2
2
3
3
4
log
2
2
3
4
1
3
4
26. log
6
3 6 log
6
6 1 3
1
3
log
6
6
1
3
1
1
3
27. log
4
16 1.2 1.2 log
4
16 1.2 log
4
4 2 1.2 2 2.4
28.
0.2 4 0.8
0.2 log
3
3 4
log
3
81 0.2 0.2 log
3
81 29. is undefined. is not in the domain of log
3
x . 9 log
3
9
30. is undefined because
is not in the domain of
log
2
x .
16
log
2
16
31. ln e
4.5
4.5
32.
12
12 1
3 ln e 4 3 4 ln e
33.
1
2
0
1
2
1
0
1
2
ln e
ln
1
e
ln 1 ln e 34.
3
4
3
4
1
3
4
ln e
ln 4 e 3 ln e 3 4
35. ln e 2 ln e 5 2 5 7
36.
7
ln e 7
ln
e
12
e 5
2 ln e
6
ln e
5
ln e
12
ln e
5
37.
2
2 log
5
5
log
5
5
2
log
5
25
log
5
75 log
5
3 log
5
75
3
38.
3
1
2
1
5
2
1
1
2
log
4
4
5
2
log
4
4
log
4
2 log
4
32 log
4
4
1 2
log
4
4
5 2 39. log
4
5 x log
4
5 log
4
x

Section 3.3 Properties of Logarithms 283
55.
4 ln x
1
2
ln y 5 ln z
ln x
4
ln y ln z
5
ln
x
4
y
z 5
ln x 4 y ln z 5
56.