Does $f(n)=\theta(g(\log{n}))$ imply $f(n)=\theta(\log{g(n)})$? I am learning about algorithmic complexities and I have this claim which I need to prove or disprove: $f(n)$ and $g(n)$ are asymptotically positive functions, if $f(n)=\Theta(g(\log(n))$ then $f(n)=\Theta(\log(g(n)))$.
Is it true?
An obvious counter example is $g(n) = \sqrt{n}$. If $f(n) = \Theta(\sqrt{\log n})$, you can't conclude that $f(n) = \Theta(\log{\sqrt{n}}) = \Theta(\log{n})$.