Logarithm problems

log₁₀x = 7
? x = 10⁷

? p^x = q
? log_pp^x = log_pq
? xlog_pp = log_pq
? x = log_pq (?log_pp = 1)

log₁₀5 = log₁₀(10/2)
= log₁₀10 - log₁₀2
= 1-0.3010
= 0.6990

log₁₀(10³.10x) = log₁₀ 10³ + log₁₀(10x)
= 3 + 2.7532
= 5.7532

If log 125 / log 5 = x
then x = 3log5 / log5= 3

(log₅3) x log₃ 5⁴ = [log3 / log5] x 4 x [log5 / log3]
= 4

Given Exp. = log₉27 - log₂₇9
= log27 / log9 - log9 / log27
= 3log3 / 2log3 - 2log3 / 3log3
= 3/2 - 2/3 = 5/6

? 10^x = 1730/1000
? log10^x= log₁₀1730 - log₁₀1000
? x = 3.2380 - 3
= 0.2380

log_an / log_abn = [( log n / log a) / (log n / log (a.b))]
= log (a.b) / log a
= ( log a + log b) / log a
= 1 + (log b / log a)
= 1 + log_ab

A = log₂₇625 + 7log₁₁¹³
= log_3³5⁴ + 7 log₁₁13
= 4/3 log₃ 5 + 7 log₁₁13
B = log₉125 + 13 log₁₁7 = log₃² 5³ + 13 log₁₁⁷
= 3/2 log₃ 5 +13 log₁₁ 7
Let log₃ 5 = x and by the above rule
7 log₁₁ 13 = 13 log₁₁ 7
Therefore, A = 4/3 x + 13 log₁₁ 7
and B = 3/2 x + 13 log₁₁ 17
clearly, A < B hence (B) is the correct answer.