2 2 log 5 5 log 5 5 2 log 5 25 log 5 75 log 5 3 log 5

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
40. log
3
10 z log
3
10 log
3
z 41. log
8
x 4 4 log
8
x 42. log
y
2
log y log 2
43.
1 log
5
x
log
5
5
x
log
5
5 log
5
x 44. log
6
z 3 3 log
6
z 45. ln z ln z 1 2
1
2
ln z
49.
ln z 2 ln z 1 , z
>
1
ln z z 1 2 ln z ln z 1 2 50.
ln x 1 ln x 1 3 ln x
ln x 1 x 1 ln x
3
ln
x 2 1
x
3
ln x 2 1 ln x 3
46. ln 3 t ln t 1 3
1
3
ln t 47.
ln x ln y 2 ln z
ln xyz 2 ln x ln y ln z 2 48.
log 4 2 log x log y
log 4 x 2 y log 4 log x 2 log y
51.
1
2
log
2
a 1 2 log
2
3, a > 1
1
2
log
2
a 1 log
2
3 2
log
2
a 1
9
log
2
a 1 log
2
9 52.
ln 6
1
2
ln x 2 1
ln 6 ln x 2 1 1 2
ln
6
x 2 1
ln 6 ln x
2
1

480 Chapter 5 Exponential and Logarithmic Functions
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.
1
2
log
2
x 4 log
2
y 4 log
2
z
log
2
x log
2
y 4 log
2
z 4
log
2
x y
4
z 4
log
2
x y 4 log
2
z 4
53.
1
3
ln x
1
3
ln y
1
3
ln x ln y
ln 3
x
y
1
3
ln
x
y
54.
ln x
3
2
ln y
1
2
2 ln x 3 ln y
1
2
ln x
2
ln y
3
ln
x 2
y 3
ln
x 2
y 3
1 2 1
2
ln
x 2
y 3
57.
2 log
5
x 2 log
5
y 3 log
5
z
log
5
x 2 log
5
y 2 log
5
z 3
log
5
x
2
y
2
z
3
log
5
x 2 log
5
y 2 z 3
58.
log x 4 log y 5 log z
log x log y
4
log z
5
log
xy 4
z
5
log xy 4 log z 5
59.
3
4
ln x
1
4
ln x 2 3
1
4
3 ln x ln x 2 3
1
4
ln x 3 ln x 2 3
ln 4 x 3 x 2 3
1
4
ln x 3 x 2 3 60.
ln x
1
2
ln x 2
ln x ln x 2 1 2
ln x x 2 1 2
ln x 2 x 2 ln x 2 x 2 1 2
61. ln x ln 3 ln 3 x
62. ln y ln t ln yt ln ty
63. log
4
z log
4
y log
4
z
y
64. log
5
8 log
5
t log
5
8
t
65. 2 log
2
x 4 log
2
x 4 2 66.
2
3
log
7
z 2 log
7
z 2 2 3
67.
1
4
log
3
5 x log
3
5 x 1 4 log
3
4 5 x
68. 4 log
6
2 x log
6
2 x
4
log
6
1
16 x 4
69.
ln
x
x 1 3
ln x 3 ln x 1 ln x ln x 1
3 70.
ln 64 z 4
5
ln 64 ln z 4
5
2 ln 8 5 ln z 4 ln 8 2 ln z 4 5
71.
log
xz 3
y 2
log
x
y 2
log z
3
log x 2 log y 3 log z log x log y 2 log z 3 72.
log
3
x
3
y
4
z 4
log
3
x 3 y 4 log
3
z 4
3 log
3
x 4 log
3
y 4 log
3
z log
3
x
3
log
3
y
4
log
3
z
4

Section 5.3 Properties of Logarithms 481
73.
ln
x
x 2 4 4
ln x ln x 2 4 4
ln x 4 ln x 2 4
ln x 4 ln x 2 ln x 2 ln x 4 ln x 2 x 2
74.
ln
z
4
z 5
4
z 5 2
ln z z 5 4 ln z 5 2
4 ln z ln z 5 2 ln z 5 4 ln z z 5 ln z 5 2
75.
ln 3
x x 3 2
x 2 1
1
3
ln
x x 3 2
x 2 1
1
3
ln x x 3 2 ln x 2 1
1
3
2 ln x 3 ln x ln x 2 1
1
3
ln x 3 2 ln x ln x 2 1
76.
ln
x 3
x 2 1
2
2 ln
x 3
x 2 1
2 ln x 3 ln x 1 x 1
2 ln x 3 ln x 1 ln x 1
2 3 ln x ln x 1 ln x 1 2 ln x
3
ln x 1 ln x 1
77.
log
8
3 y y 4
2
y 1
log
8
3
y y 4 2
log
8
y 1
1
3
log
8
y y 4 2 log
8
y 1
1
3
log
8
y 2 log
8
y 4 log
8
y 1
1
3
log
8
y log
8
y 4 2 log
8
y 1
78.
log
4
x
6
x 1 x 1
log
4
x 1 x 1 log
4
x
6
1
2
log
4
x 1 x 1
2
log
4
x
6
1
2
log
4
x 1 2 log
4
x 1 6 log
4
x
1
2
log
4
x 1 log
4
x 1
2
log
4
x
6
79.
The second and third expressions are equal by Property 2.