$2^{\log_{10}n} = n^{\log_{10}2}$ How are those equal?

Question

How are they equal? And if given $2^{\log_{10}n}$, what are the steps to convert to $n^{\log_{10}2}$?

Answer 1

Take the $\log_{10}$ of both sides.

$\log_{10}(2^{\log_{10}n}) = \log_{10}n\cdot \log_{10}2$

and similarly the right side

note, $\log_{10}$ is a strict monotonic on $(0, \infty)$.