(log n)^k = O(n)? For k greater 1

Question

$$(\log n)^k = O(n)?$$ For $k> 1$.

$k$ is a constant, such as number $4$.

I think it is not true for $n=32$ and greater. $n=32, n=64, n=128,\dots$ So, I can not find $n_0$ and $c$.

Answer 1

Here's one way: use the definition of a limit and what easily follows from l'Hopital's theorem:

$$ \lim \frac{\log^k n}{n} \to 0.$$