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4. Solve log 2 8x = 2 log 8 .
2
4
3
Solution:Since log 4 4x = 4x log 4 = 4x and 2 log 8 = 2 log 2 3 = 2 = 8
2
2
4
4
We have 4x = 8 ⇒ x = 2
a + b 1
2
2
5. If a + b = 7ab, show that log = (log a + log b).
3 2
Solution:
2
a + b a + b
2 log = log
3 3
2
2
a + b + 2ab
= log
9
= log ab
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= log a + log b
a 2 b 2 c 2
6. Prove log + log + log = 0.
bc ca ab
Solution:
2 2 2
a 2 b 2 c 2 a b c
log + log + log = log
2 2 2
bc ca ab a b c
= 0
16 25 81
7. Prove that log 2 + 16 log + 12 log + 7 log = 1.
15 24 80
Solution:
16 12 7
16 25 81 16 25 81
log 2 + 16 log + 12 log + 7 log = log 2 + log × ×
15 24 80 15 16 24 12 80 7
64 24 28
2 × 5 × 3
= log 2 + log
16
16
12
28
36
3 × 5 × 3 × 2 × 2 × 5 7
64 24 28
2 × 5 × 3
= log 2 + log
23
28
3 × 5 × 2 64
= log 2 + log 5
= log 10
= 1
8. Prove log 2 a log 2 b log 2 c = 18.
c
b
a
Solution:
2
2
2
(2 log 2 a) (2 log 2 b) (2 log 2 c) = (log 2 a ) (2 log 2 b ) (log 2 c )
b
a
c
b
a
c
n(n + 1)
n
3
2
9. Prove log a + log a + log a + · · · + log a = log a.
2
Solution:
n
2
3
log a + log a + log a + · · · + log a = log a + 2 log a + 3 log a + · · · + n log a
= (1 + 2 + 3 + · · · + n) log a

(n)(n + 1)
= log a
2