Does $ \log(x)^{x^a}$ eventually dominate $x^k$?

Question

Does $ \log(x)^{x^a}$ eventually dominate $x^k$ for all $a\gt 0$ and for all positive integers $k$?

And if so, how does one prove this? Thanks a lot for your help.

Answer 1

$$\begin{align} \log\bigl((\log x)^{x^a}\bigr)&=x^a\log\log x\\ \log(x^k)&=k\log x \end{align}$$ For all $a,k>0$$$ \lim_{x\to\infty}\frac{x^a\log\log x}{k\log x}=\infty. $$