Order of growth for algorithms: $\log(n)^{4}$or $\log(n)^{4}$ vs $5\sqrt(n)$

Question

I am not sure how to compare behaviour of $\log^4n$ or $\log^2n$ to $5\sqrt n $ for growth. Can someone help me compare it with an explanation?

Answer 1

Try taking the logarithm. $$\log(\log^4 n) =4\log \log n$$ $$\log 5\sqrt n=\frac12\log n+\log 5$$ If you are comfortable with the fact that $\log x$ is asymptotically smaller than $ax$ for any $a>0$, it is clear which is larger.