List of top Mathematics Questions on Logarithmic Differentiation asked in KCET

The differential coefficient of $\log_{10} x$ with respect to $\log_{x} 10$ is

The differential coefficient of $f (Sin \,x)$ w.r.t. $x$ where $f (x) = \log\, x$ is

Which one of the following observations is correct for the features of logarithm function to any base \(b>1\)?