I need to decide which of the following is correct:
$n^{10\log(n)} = O((\log^n(n))$
$n^{10\log(n)} = \Theta((\log^n(n))$
$n^{10\log(n)} = \Omega((\log^n(n))$
So I'm saying $n^{10\log(n)} = O((\log^n(n))$ because:
$$n^{10\log(n)} = 10^{\log(n^{10\log(n)})}=(10^{10})^{\log^2(n)}\leq c\cdot (\log^n(n)) $$
for $c=1$ and $n \geq 10^{100}$
Plus, $$\lim_{n\to \infty}\frac{n^{10\log(n)}}{\log^n(n)}=0 $$
which disprove the other two.
Is that correct? did I miss anything?