[Math] Which of $\log{\sqrt{n\log{n}}}$ and $\sqrt{\log{n}}$ grows faster

Question

Which of the following functions grows faster:

$\log{\sqrt{n\log{n}}}$ or $\sqrt{\log{n}}$?

I feel the second one should be the answer, but I find it difficult to prove as the derivatives get very complex. Does anyone know any idea?

Answer 1

The first is (ignoring very small $n$) greater than $\log (n^{1/2})= \frac12 \log n$.

So setting $a = \log n$ the first is greater than $a/2$ while the latter is $\sqrt{a}$. Whence your intuition was not correct and the first grows faster.