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?

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    $\begingroup$ No. I'm sure you can find a counterexample if you try a few functions. $\endgroup$ Commented Apr 6, 2020 at 18:41

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


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