Prove that $\log^\alpha n = o( a^n )$

Question

Please, how to prove:
$\forall c \in \mathbb R_+$ $\exists n_0 \in \mathbb N_+$ $\forall n \ge n_0 :$ $log^\alpha n < c \cdot a^n$ for $ \\a>1$, $\alpha \in \mathbb R$ ?
Thanks

Answer 1

Using the L'Hopital's theorem we have for $a>1$

$$\lim_{n\to\infty}\frac{\ln n}{a^{\frac n\alpha}}=\lim_{n\to\infty}\frac{\alpha}{n\ln a\cdot a^{\frac n\alpha}}=0$$ so for $\epsilon >0$ there's $n_0\in\Bbb N$ such that for $n\ge n_0$ we have

$$\frac{\ln n}{a^{\frac n\alpha}}<\epsilon\iff \ln^\alpha n<\epsilon ^\alpha a^n=c \cdot a^n$$