CAT 2020 Slot 3 QA Question & Solution

AlgebraMedium

If $\log_{a}{30}=A,\log_{a}({\frac{5}{3}})=-B$ and $\log_2{a}=\frac{1}{3}$, then $\log_3{a}$ equals

$\frac{2}{A+B-3}$

$\frac{2}{A+B}-3$

$\frac{A+B}{2}-3$

$\frac{A+B-3}{2}$

$\log_a30=A\ or\ \log_a5+\log_a2+\log_a3=A$...........(1)

$\log_a\left(\frac{5}{3}\right)=-B\ or\ \log_a3-\log_a5=B$.............(2)

and finally $\log_a2=3$

Substituting this in (1) we get $\log_a5+\log_a3=A-3$

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

$\log_a3=\frac{\left(A+B-3\right)}{2\ }or\ \log_3a=\frac{2}{A+B-3}$