$(\log n)^{\log n}$ lower-bound and upper-bound we know that $n \geq \log{n}$ however I understand that $(\log n)^{\log n}$ grows faster than $n$. I have been trying to prove this however I can't seem to figure it out.
Here is a way to show it without limits. Let $n = 2^x$. Now you are comparing the growth rates of $2^x$ and $x^x$.