Convergence of $\int_0^2\frac{1}{x^\alpha |\ln^\beta(x)|}$

Question

I know that $\int_2^{+\infty}\frac{1}{x^\alpha\ln^\beta(x)}$ is
$i)$ Convergent if $\alpha\gt1$, $\forall\beta$
$ii)$Convergent if $\beta\gt1$ for $\alpha=1$
$iii)$Divegent if $\alpha\lt1$,$\forall\beta$
Now i am trying to understand what happens if i have same situation with interval other interval,say $(0,2)$.Can something similar be formulated for $\int_0^2\frac{1}{x^\alpha\ |ln^\beta(x)|}$?