For all (fixed) positive real numbers $\alpha > 0$, we have that $$\lim_{x\to\infty}\frac {\ln(x)}{x^\alpha}=0,$$ $$\lim_{x\to\infty}\frac {x^\alpha}{\ln (x)}=\infty,$$ $$\lim_{x\searrow 0}{x^\alpha}{\ln (x)}=0.$$
These limits show that if $x$ is growing, the growth of the logarithmic function of $x$ is slower than the growth of $x$ to the power of $\alpha$, no matter small $\alpha$ is.
Proofs: 1
