Which grows faster, $n 2^{\sqrt{\log n}}$ or $n^{4/3}$?

Question

Which one grow faster $n 2^{\sqrt{\log n}}$ or $n^{4/3}$? The log base is 2. My mind tells me first one is grow faster because we have $n$ at the exponential position even though it comes with $\log$ and $\sqrt{.}$. But when I test a large case on computer, computer tells me the second one is larger. Any suggestions?

Answer 1

Taking the suggestions in the comments to heart, let's divide by $n$ and the the logarithm (base $2$), giving:

$$\sqrt{\log n } \quad \wedge \quad \frac{1}{3}\log n,$$

so the second expression will clearly grow faster. It overtakes the first when $$\sqrt{\log n }=\frac{1}{3}\log n \implies n=2^9,$$

where we have thrown away the solution $n=1$, as it clearly doesn't fit our criterion.

Answer 2

Taking $\log_n$ on both sides gives $$1 + \frac{1}{\sqrt{\log_2 n}}$$ for the first one and $$4/3$$ for the second one. So once $n > 2^9$, the second expression is larger.