CAT (QA) PYQs

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

If $x$ is a positive real number such that $4 \log _{10} x+4 \log _{100} x+8 \log _{1000} x=13$, t hen greatest integer not exceeding x , is

$\dfrac{1}{\log _{2} 100}-\dfrac{1}{\log _{4} 100}+\dfrac{1}{\log _{5} 100}-\dfrac{1}{\log _{10} 100}+\dfrac{1}{\log _{20} 100}-\dfrac{1}{\log _{25} 100}+\dfrac{1}{\log _{50} 100}= \ ?$