Does $f(n)=\theta(g(\log{n}))$ imply $f(n)=\theta(\log{g(n)})$?
An obvious counter example is $g(n) = \sqrt{n}$. If $f(n) = \Theta(\sqrt{\log n})$, you can't conclude that $f(n) = \Theta(\log{\sqrt{n}}) = \Theta(\log{n})$.