How to compare power of logarithm function $\log x$ with the power of $x$?

Question

Let $r_1, r_2 \in \mathbb R$. Assume that $r_2<-2.$Can we find generic relation between $r_1$ and $r_2$ so that

$$ (\log N)^{r_1} N^{r_2} \ll N^{-2}? $$

Or What condition on $r_1$ ensure the above inequality?

Answer 1

The relation is simply $r_2 < -2$; for any $\epsilon > 0$, $\log(N) = \mathcal O(N^\epsilon)$, so also $\log(N)^{r_1} = \mathcal O(N^{\epsilon r_1})$.