Limiting behavior of $n^a$ and $\log_{b}{n}$, $n\in\mathbb{N}$

Question

Let $\{u_n\}_{n\in\mathbb{N}}=n^a$ and $\{v_n\}_{n\in\mathbb{N}}=\log_{b}{n}$ be two sequences where $a>1$ and $b>1$. Then it's known that $$v_n\in o\left(u_n\right)$$ If we take $0<a\leq 1$ is above relation true?

Answer 1

$$\lim\frac{n^a}{\log_b n}=\lim\frac{n^a\ln b}{\ln n}=\lim_{x\to\infty}\frac{x^a\ln b}{\ln x}\stackrel*=\lim_{x\to\infty}ax^a\ln b$$

(*) l'Hopital's rule has been applied here.