For which $\alpha, \beta \in \mathbb{R}$ is $\frac{1}{x^\alpha (\log(x))^\beta} \in L^p((1,\infty)$ where $1\leq p \leq \infty$

Question

I'm having a really hard time finding a starting point here. I think that it won't work for any $\alpha, \beta >0$ but I am not even sure about that.

Answer 1

You have to study the integral $$ \int_0^\infty\frac{dx}{x^{p\alpha}(\log x)^{p\beta}}. $$ At $x=1$, $\log x\sim x-1$. This will give you an integrability condition on $\beta$. At $x=\infty$, the dominant term is $x^{-p\alpha}$, and you get an integrability condition on $\alpha$.