CAT (QA) PYQs

Suppose, $\log_3 x = \log_{12} y = a$, where $x, y$ are positive numbers. If $G$ is the geometric mean of x and y, and $\log_6 G$ is equal to

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