PROBLEM & SOLUTION -
a³b⁴
Example.1) Express log₁₀ ------- in terms of log₁₀ a, log₁₀ b, and log₁₀ c
√c
a³b⁴
Ans.) We have log₁₀ ---------
√c
=> log₁₀ (a³b⁴) – log₁₀ (√c) [using Quotient Law]
=> log₁₀ a³ + log₁₀ b⁴ - log₁₀ (c)⅟² [using Product Law]
=> 3 log₁₀ a + 4 log₁₀ b – 1/2 log₁₀ c [using Power Law] (Ans.)
Example.2) log 5 + log 20 + log 24 + log 25 – log 60
Ans.) log 5 + log 20 + log 24 + log 25 – log 60
=> log (5 X 20 X 24 X 25) – log 60
5 X 20 X 24 X 25
=> log (------------------)
60
=> log (5 X 8 X 25)
=> log 1000
=> log 10³
=> 3 (Ans.)
Example.3) log 6 + 2 log 5 + log 8 – log 3 – log 4
Ans.) log 6 + 2 log 5 + log 8 – log 3 – log 4
=> log₁₀ 6 + log₁₀ 5² + log₁₀ 8 – log₁₀ 3 – log₁₀ 4
=> (log₁₀ 6 + log₁₀ 5² + log₁₀ 8) – (log₁₀ 3 + log₁₀ 4)
=> log₁₀ (6 X 25 X 8) – log₁₀ (3 X 4)
6 X 25 X 8
=> log₁₀ (-------------)
3 X 4
=> log₁₀ 100
=> log₁₀ 10²
=> 2 (Ans.)
Example.4) Show that:-
16 25 81
7 log (------) + 5 log (------) + 3 log (------) = log 2
15 24 80
16 25 81
Ans.) 7 log (-------) + 5 log (--------) + 3 log (--------)
15 24 80
16 25 81
=> log [(--------)⁷ ] + log [(---------)⁵] + log [(---------)³]
15 24 80
16 25 81
=> log [(---------)⁷ X (---------)⁵ X (---------)³]
15 24 80
2⁴ 5² 3⁴
=> log [(---------)⁷ X (---------)⁵ X (---------)³]
3 X 5 3 X 2³ 2⁴ X 5
2²⁸ 5¹⁰ 3¹²
=> log [(----------) X (-----------) X (-----------)]
3⁷ X 5⁷ 3⁵ X 2¹⁵ 2¹² X 5³
=> log [ 2²⁸ X 5¹⁰ X 3¹² X 3ˉ⁷ X 5ˉ⁷ X 3ˉ⁵ X 2ˉ¹⁵ X 2ˉ¹² X 5ˉ³ ]
=> log ( 2²⁸ˉ¹⁵ˉ¹² X 3¹²ˉ⁷ˉ⁵ X 5¹⁰ˉ⁷ˉ³ )
=> log (2²⁸ˉ²⁷ X 3¹²ˉ¹² X 5¹⁰ˉ¹⁰ )
=> log (2¹ X 3⁰ X 5⁰)
=> log (2 X 1 X 1)
=> log 2 (Proven)
log 64
Example.5) Evaluate, (i) -------------
log 8
log 64
Given, -----------
log 8
log 2⁶
=> ------------
log 2³
6 log 2
=> ----------- = 2 (Ans.)
3 log 2
log 81
Example.5) (ii) -----------
log 27
log 81
Given, ------------
log 27
log 3⁴
=> -------------
log 3³
4 log 3
=> -----------
3 log 3
4
=> ------ (Ans.)
3
1
Example.6) Express (2 log 3 - ------- log 729 + log 12)
3
1
Given, 2 log 3 - ------- log 729 + log 12
3
=> log 3² – log (3⁶)⅓ + log 12
=> log 9 – log 9 + log 12
=> log 12 (Ans.)
Example.7) If, log 2 = x, log 3 = y, and log 7 = z, express log ₁₀ {(4)3√63} in terms of x, y, and z
Ans.) log₁ {(4)3√63} = log 4 + log 3√63
= log 2²+ log (63)⅓
1
= 2 log 2 + ------ log (7 X 9)
3
1
= 2 log 2 + ------- log (7 X 3²)
3
1
= 2 log 2 + ------- [ log 7 + 2 log 3 ]
3
1 2
= 2 log 2 + ------- log 7 + ------- log 3
3 3
Now, we will substitute the value log 2, log 3, and log 7, and we get –
= 2 x + 1/3 y + 2/3 z (Ans.)