Does (or when does) $O(\log\text{poly}(n)) = O(\text{poly}(\log(n)))$?

Question

Since $P(n) = O(n^{\deg P})$, is $\log P(n) = O(P(\log n))$? Or when does this occur?

Answer 1

Hint:

$$\lim_{n \to \infty} \frac{\log P(n)}{\log(n)} = \operatorname{deg}(P) < \infty$$