asymptotic analysis of log bases

Asymptotic analysis of ${\log }_k$

$f(n)={\log }_{3}n$, $g(n)={\log }_{2}n$ Is $f(n)=O(g(n))$? Is $g(n)=O(f(n))$?

Yes to both questions, because log-bases can be treated as constant coefficients.

$${\log }_b(a)=\frac{{\log }_x(a)}{{\log }_x(b)}$$

The change-of-base formula dictates that every base is related to another base by only a constant factor.