MCQ: Logarithms - 3 - SSC CGL MCQ

∵ (1000)^x = 3
⇒ xlog 10³ = log 3
⇒ 3x = log 3
∴ x = log 3 / 3 = 0.477 / 3 = 0.159 

∵ log₂³ x¹ +log₂² x¹ + log₂x = 11
⇒ 1/3log₂x + 1/2log₂x + log₂x = 11
⇒ (1/3 + 1/2 +1)log₂x = 11
⇒ 11/6log₂x = 11
⇒ log₂x = 11 x 6 / 11 =6
∴ x = 2⁶ = 64 

Exp. = log_0.12564 = log₂^-3 2⁶
={6 / (-3)} log₂2
= -2 

∵ 5^5-x = 2^x-5
⇒ 5^5-x = 2^-(5-x)
⇒ (5-x)log 5 = -(5-x)log 2
⇒ (5-x)log 5 + (5-x)log 2 = 0
⇒ (5-x){log 5 + log 2 } = 0
⇒ (5-x){log10/2 + log 2 } = 0
⇒ (5-x){log10 - log 2 + log 2} = 0
⇒ 5-x = 0
∴ x = 5 

= log_0.125125 = log₂^-3 5³ = - 3/3log₂5 = -log₂5
∵ 10^0.3010 = 2
⇒ log₁₀2 = 0.3010
∵ log₁₀5 = log₁₀10/2 = log₁₀10-log₁₀2
= 1-0.3010 = 0.6990
∴ -log₂5 = - log₁₀⁵ / log₁₀²
= - 0.6990 / 0.3010 = - 699/301 

=log₃₂2⁸ + log₂₄₃3⁷ - log₃₆1296
= log₂⁵ 2⁸ + log₃⁵ 3⁷ - log₃₆ 36²
= (8/5)log₂2 + (7/5)log₃3 - 2log₃₆36
= 8/5 + 7/5 - 2 = 1 

log₉ 81 - log₄ 32 = log₃² 3⁴ - log₂² 2⁵
= 4/2 - 5/2 = -1/2 

log₁₀xy = z
⇒ xy = 10^z
⇒ x = 10^z/y 

= log₄₉16807 - log₉27
= log₇² 7⁵ - log₃² 3³
=(5/2)log₇7 - (3/2)log₃3
=5/2 - 3/2
= 1 

= log₁₀10 + log₁₀100 + log₁₀1000 + log₁₀10000 + log₁₀100000
= 1 + 2 + 3 + 4 + 5
= 15 

Let log_-1/381 = x
⇒ 81 = (-1/3)^x
⇒ 3⁴ = (-1/3)^x = 3^-x
∴ x = -4 

= 25 log₅4 = 5² log₅4
= 5 log₅ 4²
= 16 

= log₁₀ 10^-5 = -5

If log₁₀{log₁₀[log₁₀(log₁₀N)]} = 0
⇒ log₁₀{log₁₀(log₁₀N)} = 1
⇒ log₁₀(log₁₀N) = 10
⇒ log₁₀^N = 10¹⁰
∴ N = 10¹⁰¹⁰ 

= log₁₀ 10^-6 = -6