CAT 2023

Slot 2

QA

Question & Solution

It is given that  $\log_{\sqrt{3}}{(x)}+\frac{\log_{x}{(25)}}{\log_{x}{(0.008)}}=\frac{16}{3}$, which can be written as:

=> $2\log_3x+\log_{0.008}25\ =\ \frac{16}{3}$

=> $2\log_3x+\log_{\frac{8}{1000}}25\ =\ \frac{16}{3}$

=> $2\log_3x+\log_{\frac{1}{125}}25\ =\ \frac{16}{3}$

=> $2\log_3x+\log_{5^{-3}}\left(5\right)^2\ =\ \frac{16}{3}$

=> $2\log_3x-\frac{2}{3}=\ \frac{16}{3}$

=> $2\log_3x=\frac{16}{3}+\frac{2}{3}$

=> $2\log_3x=6$

=> $\log_3x^2=6\ =>\ x^2\ =\ 3^6$

Hence, $\log_3\left(3\cdot x^2\right)\ =\ \log_3\left(3\cdot3^6\right)\ =\log_33^7\ =\ 7$