cat 2020 Question 3

Question

Let $log_{a}$30 = A, $log_{a}\frac{5}{3}$ = -B and $log_{2}$a = $\frac{1}{3}$ , then $log_{3}$a equals

Accepted Answer

$\log_a 30 = A \ ext{ or }\ \log_a 5 + \log_a 2 + \log_a 3 = A \quad ...(1)$

$\log_a \left(\frac{5}{3} ight) = -B \ ext{ or }\ \log_a 3 - \log_a 5 = B \quad ...(2)$

$ ext{and finally } \log_a 2 = 3$

$ ext{Substituting this in (1) we get } \log_a 5 + \log_a 3 = A - 3$

$ ext{Now we have two equations in two variables (1) and (2). On solving we get}$

$\log_a 3 = \frac{A + B - 3}{2} \quad ext{ or }\ \log_3 a = \frac{2}{A + B - 3}$

Question 3.