\begin{align}

0 \lt \frac1t & \lt t^{c-1} &(t \gt 1, \, c \gt 0) \\

0 \lt \int_1^x \frac1t \,\mathrm dt & \lt \int_1^x t^{c-1} \,\mathrm dt \\

0 \lt \log t &\lt \frac1c ( x^c -1) \lt \frac{x^c}c \\

0 \lt \frac{\log^b x}{x^a} \lt \frac {x^{-a}}{c^b} &(a \gt 0, \, b \gt 0) \\

\text{(choose $c= \tfrac{a}{2b}$ and} \; 0 \lt \frac{\log^b x}{x^a} \lt \frac {x^{-\frac{a}2}}{c^b}

\end{align}

and take limits as $x \to \infty$.