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1. QUESTIONS AND SOLUTIONS
LOGARITHM
Written by :
SYAHLUL ERBI SYAPUTRA

2. QUESTIONS AND SOLUTIONS
1. If log
𝑎2
𝑏2
= 12, so that log √
𝑏
𝑎
3
equal…
Solution :
log
𝑎2
𝑏2
= 12 log √
𝑏
𝑎
3
= log (
𝑏
𝑎
)
1
3
log (
𝑎
𝑏
)
2
= 12 =
1
3
log
𝑏
𝑎
2 log
𝑎
𝑏
= 12 =
1
3
(log b – log a)
log
𝑎
𝑏
= 6 = –
1
3
(log a – log b)
= –
1
3
log
𝑎
𝑏
= –
1
3
(6)
= –2
2. If 7
log 2 = a and 2
log 3 = b, so that 6
log 98 is…
Solution :
7
log 2 . 2
log 3 = ab, so that 7
log 3 = ab
6
log 98 = 6
log 2.72
= 6
log 2 + 6
log 72
= 6
log 2 + 2 6
log 7
2
log 6 = 2
log 2 . 3 = 2
log 2 + 2
log 3 = 1 + b
So that 6
log 2 =
1
1+𝑏
7
log 6 = 7
log 2 . 3 = 7
log 2 + 7
log 3 = a + ab
So that, 6
log 7 =
1
𝑎+𝑎𝑏

3. So, 6
log 98 = 6
log 2 + 2 6
log 7
=
1
1 + 𝑏
+ (
1
𝑎 + 𝑎𝑏
)
=
𝑎
𝑎(1 + 𝑏)
+
2
𝑎(1 + 𝑏)
=
𝑎 + 2
𝑎(1 + 𝑏)
3. a
log
1
𝑏
. b
log
1
𝑐2
. c
log
1
𝑎3
is…
Solution :
a
log
1
𝑏
. b
log
1
𝑐2
. c
log
1
𝑎3
= a
log 𝑏−1
. b
log 𝑐−2
. c
log 𝑎−3
= (–1 a
log b) (–2 b
log c) (–3 c
log a)
= –6 . a
log b . b
log c . c
log a
= –6 a
log a
= –6 . 1
= –6
4. If 𝑓(𝑥) =
𝑙𝑜𝑔3 𝑥
1−2 𝑙𝑜𝑔3 𝑥
, so that 𝑓(𝑥) + 𝑓 (
3
𝑥
) is…
Solution :
𝑓(𝑥) =
𝑙𝑜𝑔3 𝑥
1 − 2 𝑙𝑜𝑔3 𝑥
𝑓 (
3
𝑥
) =
𝑙𝑜𝑔3 (
3
𝑥
)
1 − 2 𝑙𝑜𝑔3 (
3
𝑥
)
=
𝑙𝑜𝑔3 3 − 𝑙𝑜𝑔3 𝑥
1 − 2(𝑙𝑜𝑔3 3 − 𝑙𝑜𝑔3 𝑥)
=
1 − 𝑙𝑜𝑔3 𝑥
1 − 2 + 2 𝑙𝑜𝑔3 𝑥
=
1 − 𝑙𝑜𝑔3 𝑥
−1 + 2 𝑙𝑜𝑔3 𝑥
=
−1 − 𝑙𝑜𝑔3 𝑥
1 − 2 𝑙𝑜𝑔3 𝑥

4. 𝑓(𝑥) + 𝑓 (
3
𝑥
) =
𝑙𝑜𝑔3 𝑥
1 − 2 𝑙𝑜𝑔3 𝑥
+
−1 − 𝑙𝑜𝑔3 𝑥
1 − 2 𝑙𝑜𝑔3 𝑥
=
−1 + 2 𝑙𝑜𝑔3 𝑥
1 − 2 𝑙𝑜𝑔3 𝑥
=
−(1 − 2 𝑙𝑜𝑔3 𝑥
1 − 2 𝑙𝑜𝑔3 𝑥
= – 1
5. 5
log √27 9
log 125 + 16
log 32 is…
Solution :
5 log √27 9 log 125 + 16 log 32
= 𝑙𝑜𝑔5 3
1
2 . 𝑙𝑜𝑔32 53
+ 𝑙𝑜𝑔24 25
=
3
2
𝑙𝑜𝑔5 3 .
3
2
𝑙𝑜𝑔3 5 +
5
4
𝑙𝑜𝑔2 2
=
9
4
5𝑙𝑜𝑔 5 +
5
4
=
9
4
+
5
4
=
14
4
=
7
2
6. If a
log 3 = b
log 27 , 𝑎 > 0, 𝑏 > 0, 𝑎 ≠ 1, 𝑏 ≠ 1, so that a
log b is…
Solution :
𝑎𝑙𝑜𝑔 3 = 𝑏𝑙𝑜𝑔 27
𝑙𝑜𝑔 3
𝑙𝑜𝑔 𝑎
=
𝑙𝑜𝑔 27
𝑙𝑜𝑔 𝑏
𝑙𝑜𝑔 3
𝑙𝑜𝑔 𝑎
=
3𝑙𝑜𝑔 3
𝑙𝑜𝑔 𝑏
𝑙𝑜𝑔 𝑏 = 3𝑙𝑜𝑔 𝑎
𝑎𝑙𝑜𝑔 𝑏 =
𝑙𝑜𝑔 𝑏
𝑙𝑜𝑔 𝑎

5. =
3𝑙𝑜𝑔 𝑎
𝑙𝑜𝑔 𝑎
= 3
7. If 9
log 8 = 3m, so that value of 4
log 3 is…
Solution :
9𝑙𝑜𝑔 8 = 3𝑚 ↔
𝑙𝑜𝑔 8
𝑙𝑜𝑔 9
= 3𝑚
↔
𝑙𝑜𝑔 23
𝑙𝑜𝑔 32
= 3𝑚
↔ 3 𝑙𝑜𝑔 2 = 3𝑚 . 2 𝑙𝑜𝑔 3
↔ 𝑙𝑜𝑔 2 = 2𝑚 𝑙𝑜𝑔 3
4𝑙𝑜𝑔 3 =
𝑙𝑜𝑔 3
𝑙𝑜𝑔 22
=
𝑙𝑜𝑔 3
2 𝑙𝑜𝑔 2
=
𝑙𝑜𝑔 3
2.2𝑚 𝑙𝑜𝑔 3
=
1
4𝑚
8. If 8
log b = 2 and 4
log d = 1, the relationship between the values of b and d is …
Solution :
relation (1) and (2) is:
log3 𝑏
log4 𝑑
=
2
1
8
log b = 2 .4
log d
log 𝑏
log 8
= 2 .
log 𝑑
log 4
log 𝑏
log 23
= 2 .
log 𝑑
log 22
log 𝑏
3log 2
= 2 .
log 𝑑
2log 2
log 𝑏
3log 2
=
log 𝑑
log 2
1
3
2
log b = 2
log d
log2 𝑏
1
3 = 2
log d
𝑏
1
3 = d, so that relation b and d is b = 𝑑3

6. 9. (2log2 6
)(3log9 5
)(5log𝑎 2
) with a =
1
5
, equal…
Solution :
(2log2 6
)(3log9 5
)(5log𝑎 2
)
= (2log2 6
) . (3
1
2
log3 5
) . (5−1 log5 2
)
= (2log2 6
) . (3log3 √5) . (5log5 2−1
)
= 6 . √5 . 2−1
= 3√5
10. If x > 1, so that
1
log𝑚 𝑥
+
1
log𝑛 𝑥
equal…
Solution :
1
log𝑚 𝑥
+
1
log𝑛 𝑥
= x
log m + x
log n
= x
log (m.n)
= x
log mn
11. The value from
3+log(log 𝑥)
log3(log 𝑥1000)
is…
Solution :
3 + 𝑙𝑜𝑔(𝑙𝑜𝑔 𝑥)
𝑙𝑜𝑔3(𝑙𝑜𝑔 𝑥1000)
=
𝑙𝑜𝑔 1000 + 𝑙𝑜𝑔(𝑙𝑜𝑔 𝑥)
𝑙𝑜𝑔3( 1000 . 𝑙𝑜𝑔 𝑥)
=
𝑙𝑜𝑔(1000 . 𝑙𝑜𝑔 𝑥)
𝑙𝑜𝑔(1000 . 𝑙𝑜𝑔 𝑥)
𝑙𝑜𝑔 3
= 𝑙𝑜𝑔 3 ×
𝑙𝑜𝑔(1000. 𝑙𝑜𝑔 𝑥)
𝑙𝑜𝑔(1000. 𝑙𝑜𝑔 𝑥)
= 𝑙𝑜𝑔 3

7. 12. The value of x that satisfies: log x = 4 (a + b)+ 2 log (a – b) – 3 log (𝑎2
− 𝑏2
) – log
𝑎+𝑏
𝑎−𝑏
is...
Solution :
𝑙𝑜𝑔 𝑎 – 𝑙𝑜𝑔 𝑏 = 𝑙𝑜𝑔
𝑎
𝑏
𝑙𝑜𝑔 𝑎 + 𝑙𝑜𝑔 𝑏 = 𝑙𝑜𝑔 𝑎𝑏
𝑙𝑜𝑔 𝑥 = 4 (𝑎 + 𝑏) + 2 𝑙𝑜𝑔 (𝑎 – 𝑏) – 3 𝑙𝑜𝑔 (𝑎2
− 𝑏2
) – 𝑙𝑜𝑔
𝑎 + 𝑏
𝑎 − 𝑏
𝑙𝑜𝑔 𝑥 =
𝑙𝑜𝑔(𝑎 + 𝑏)4(𝑎−𝑏)2
(𝑎2 − 𝑏2)3
(𝑎 − 𝑏)
𝑙𝑜𝑔 𝑥 =
𝑙𝑜𝑔{(𝑎 + 𝑏)}4(𝑎−𝑏)2
(𝑎2 − 𝑏2)3(𝑎 + 𝑏)
𝑙𝑜𝑔 𝑥 = 𝑙𝑜𝑔 1
𝑥 = 1
13. If a
log b = x and b
log d = y, so that d
log a is expressed in x and y is…
Solution :
𝑎𝑙𝑜𝑔 𝑏 = 𝑥
𝑙𝑜𝑔 𝑏
𝑙𝑜𝑔 𝑎
= 𝑥
𝑏𝑙𝑜𝑔 𝑑 = 𝑦
𝑙𝑜𝑔 𝑑
𝑙𝑜𝑔 𝑏
= 𝑦
𝑙𝑜𝑔 𝑏
𝑙𝑜𝑔 𝑎
×
𝑙𝑜𝑔 𝑑
𝑙𝑜𝑔 𝑏
= 𝑥𝑦
𝑙𝑜𝑔 𝑑
𝑙𝑜𝑔 𝑎
= 𝑥𝑦
𝑙𝑜𝑔 𝑎
𝑙𝑜𝑔 𝑑
=
1
𝑥𝑦
𝑑𝑙𝑜𝑔 𝑎 =
1
𝑥𝑦

8. 14. If x
log 625 = 4 and 8
log y = 0, so that x – y is…
Solution :
x
log 625 = 4 8
log y = 0 So, x – y = 5 – 1 = 4
𝑥4
= 625 80
= 𝑦
𝑥4
= 54
y = 1
x = 5
15. If 9𝑙𝑜𝑔 𝑎 = – 1 and log1
𝑎
𝑥 =
1
2
, so that value of x is…
Solution :
1
2
3𝑙𝑜𝑔 𝑎 = – 1
3𝑙𝑜𝑔 𝑎 = – 1
𝑎 = 3−2
𝑎 =
1
9
𝑙𝑜𝑔1
𝑎
𝑥 =
1
2
↔ 9𝑙𝑜𝑔 𝑥 =
1
2
𝑥 = 9
1
2
𝑥 = √9
𝑥 = 3
16. If 6(340)(log2 log 𝑎) + (log2 log 𝑎) so that value of a is…
Solution :
6(340)(log2 log 𝑎) + (log2 log 𝑎)
6 (340)(log2 𝑎) + 341(log2 𝑎) = 343
341
. [2. log2 𝑎 + log2 𝑎] = 343
341
. [3. log2 log2 𝑎] = 343
log2 𝑎 = 3
a = 23
a = 8

9. 17. If
log3 𝑥
log3 𝑤
= 2 and xy
log w =
2
5
, so that value of
log2 𝑤
log2 𝑦
is…
Solution :
xy
log w =
2
5
↔ w
log xy =
5
2
w
log x + w
log y =
5
2
2 + w
log y =
5
2
w
log y =
1
2
→ w
log y =2
log3 𝑥
log3 𝑤
= 2
w
log x = 2
18. If f(x) = log (
1+𝑥
1−𝑥
) , so that f (
𝑥1+𝑥₂
1+𝑥₁𝑥₂
)
Solution :
𝑓 (
𝑥1
+ 𝑥₂
1 + 𝑥₁𝑥₂
) = 𝑙𝑜𝑔 (
1 +
𝑥1
+ 𝑥₂
1 + 𝑥₁𝑥₂
1 −
𝑥1 + 𝑥₂
1 + 𝑥₁𝑥₂
)
= 𝑙𝑜𝑔 (
𝑥₁𝑥₂ + 1 + 𝑥₁ + 𝑥₂
𝑥₁𝑥₂ + 1 − 𝑥₁ − 𝑥₂
)
= 𝑙𝑜𝑔 (
𝑥₁𝑥₂ + 1 + 𝑥₁ + 𝑥₂
𝑥₁𝑥₂ + 1 − 𝑥₁ − 𝑥₂
)
= 𝑙𝑜𝑔 (
1 + 𝑥₁
1 − 𝑥₁
) + 𝑙𝑜𝑔 𝑙𝑜𝑔 (
1 + 𝑥₂
1 − 𝑥₂
)
= 𝑙𝑜𝑔 (
1 + 𝑥₁
1 − 𝑥₁
) + 𝑙𝑜𝑔 𝑙𝑜𝑔 (
1 + 𝑥₂
1 − 𝑥₂
)

10. 19. If b
log a +log√5 𝑎 = 4, so that a
log b is…
Solution :
b
log a +log√5 𝑎 = 4
b
log log a + 2 b
log log a = 4
3 b
log log a = 4
b
log log a = 4
a
log log b =
3
4
20. If p
log q – q
log p = 6, so that value of (log𝑝 𝑞)
2
+ (log𝑞 𝑝)
2
is…
Solution :
(log𝑝 𝑞 − log𝑞 𝑝)
2
= 62
(log𝑝 𝑞)
2
− 2 log𝑝 𝑞 log𝑞 𝑝 + (log𝑞 𝑝)
2
(log𝑝 𝑞)
2
+ (log𝑞 𝑝)
2
= 36 + 2
(log𝑝 𝑞)
2
−2.1 + (log𝑞 𝑝)
2
= 36
(log𝑝 𝑞)
2
+ (log𝑞 𝑝)
2
= 38