CAT (QA) PYQs

If $\log (2^a \times 3^b \times 5^c)$ is the arithmetic mean of $\log (2^2 \times 3^3 \times 5)$, $\log (2^6 \times 3 \times 5^7)$, and $\log (2 \times 3^2 \times 5^4)$, then a 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

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 $\log_3(3x^2)$ is