I was wondering if somebody could help me this homework question.
Find all $p, q > 0$ with the following properties: (a) if $0 \leq f \in L_1(0, 1)$, then $f^p \log^q(f) \in L_1(0, 1)$.
This is for a measure theory course (graduate level). I have tried using Jensen's inequality, Holder's inequality, bounding $\log(f(x))$ by some other function, and also I've tried using a substitution $x = e^t$ and solving the integrals. Any help would be greatly appreciated.