For what conditions of $a$ and $b$, it follows that $ \lim_{x \to \infty} \frac{\ln^a x}{x^b}=0 $?

Question

So the elemental limit

$$ \lim_{x \to \infty} \frac{\ln x}{x}=0 $$

Makes it possible to know that for any $b \ge1$ the following limit follows: $$ \lim_{x \to \infty} \frac{\ln x}{x^b}=0 $$

However, what are the universal conditions for $a$, $b$ such that the limit $$ \lim_{x \to \infty} \frac{\ln^a x}{x^b}=0 $$ Holds true?

Thanks in advance!

Answer 1

$b > 0$, no condition on $a$. Consider that $$ \frac{\ln^a x}{x^b} = \left( \frac{\ln^{a/b} x}{x} \right)^b = \left( \frac{\ln x}{x^{b/a}} \right)^a \text{.} $$