When does $\int_2^\infty \frac{1}{x^p \log^q x} dx$ converge?

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What are the values of $p$, $q$ for which an improper below is convergent?:

$$\int_2^\infty \frac{1}{x^p \log^q x}dx$$

I divided the case and did a comparison test, substitution. But I want an easier way.

Do you have some idea?