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?

For a real number $a$, if $\frac{\log _{15} a+\log _{32} a}{\left(\log _{15} a\right)\left(\log _{32} a\right)}=4$ then a must lie in the range.

The sum of all distinct real values of $x$ that satisfy the equation $10^x + \frac{4}{10^x} = \frac{81}{2}$, is