CAT 2023 Slot 2 QA Q5 Logarithms & Exponents

Question 5 of the Quantitative Ability (QA) section, CAT 2023 Slot 2, with CAT's official answer and a worked solution.

For some positive real number \(x\), if \(\log _{\sqrt{3}}(x)+\frac{\log _x(25)}{\log _x(0.008)}=\frac{16}{3}\), then the value of \(\log _3\left(3 x^2\right)\) is

TITA: type in the answer. There are no options and no negative marking.

Answer: 7 Logarithms & Exponents TITA

Solution

\(\log_{\sqrt3} x = 2\log_3 x\).
\(\frac{\log_x 25}{\log_x 0.008} = \frac{\log 25}{\log 0.008} = \frac{2\log 5}{-3\log 5} = -\frac{2}{3}\), since \(0.008 = 5^{-3}\).
So \(2\log_3 x - \frac23 = \frac{16}{3}\), giving \(\log_3 x = 3\).
\(\log_3(3x^2) = 1 + 2\log_3 x = 1 + 6 = 7\).