# Testing convergence of improper integral with a variable a

I have trouble determining for which $$a\in \mathbb{R}\;$$ the following improper integral converges:
$$\int_0^1\frac{\ln(x)}{x^a}dx$$ I have tried the following:
$$\left|\frac{\ln(x)}{x^a}\right|=\frac{-\ln(x)}{x^a}$$ for $$0
and: $$\frac{1}{x}>-\ln(x)$$ on this interval.
So $$\frac{\frac{1}{x}}{x^a}>\frac{-\ln(x)}{x^a}$$ (on this interval)
$$\frac{\frac{1}{x}}{x^a}$$ can be written as a p-integral: $$\int_0^1 x^{-p} dx$$ with $$p = a+1.$$ Thus, I concluded, the improper integral converges for $$a<0$$ since the p-integral does too.

However, I'm asking myself whether this is the right way to tackle these kinds of problems, especially since I have no idea how to prove/disprove convergence for $$a\geq 0.$$ Is what I have done above right? And how should I go about testing convergence for $$a\geq 0$$?