CAT (QA) PYQs

Let $x$ and $y$ be positive real numbers such that $\log _{5}(x+y)+\log _{5}(x-y)=3$, and $\log _{2} y-\log _{2} x=1-\log _{2} 3$. Then $x y$ 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

The number of distinct integer values of n satisfying $\frac{4-\log_2 n}{3-\log_4 n} < 0$, is