CAT (QA) PYQs

If $x$ is a real number such that $\log_3 5 = \log_5 (2 + x)$, then which of the following is true?

If $\log _{4} 5=\left(\log _{4} y\right)\left(\log _{6} \sqrt{ } 5\right)$, then $y$ equals

For a real number $x$, if $\frac{1}{2}, \frac{\log _{3}\left(2^{x}-9\right)}{\log _{3} 4}$, and $\frac{\log _{5}\left(2^{x}+\frac{17}{2}\right)}{\log _{5} 4}$ are in an arithmetic progression, then the common difference is