CAT 2020 Slot 3 QA Question 12: If $\log_a 30 = A$, $\log_a \dfrac{5}{3} = -B$ and $\log_2 a = \dfrac{1}{3}$, then $\log_3 a$ equals

Question

If $\log_a 30 = A$, $\log_a \dfrac{5}{3} = -B$ and $\log_2 a = \dfrac{1}{3}$, then $\log_3 a$ equals

Accepted Answer

Let $x = \log_a 3$ and $y = \log_a 5$.
Since $\log_2 a = \dfrac{1}{3}$, we have $\log_a 2 = 3$.
$\log_a 30 = \log_a 2 + \log_a 3 + \log_a 5 = A$
$\Rightarrow 3 + x + y = A$
$\Rightarrow x + y = A - 3$ ---- (1)
$\log_a \dfrac{5}{3} = y - x = -B$
$\Rightarrow x - y = B$ ---- (2)
Adding (1) and (2),
$2

If log⁡a30=A\log_a 30 = Aloga​30=A, log⁡a53=−B\log_a \dfrac{5}{3} = -Bloga​35​=−B and log⁡2a=13\log_2 a = \dfrac{1}{3}log2​a=31​, then log⁡3a\log_3 alog3​a equals

If $\log_a 30 = A$ , $\log_a \dfrac{5}{3} = -B$ and $\log_2 a = \dfrac{1}{3}$ , then $\log_3 a$ equals