coniditons for integrability for log(X) where X is a r.v.?

Question

Let $X$ be a positive random variable. I was thinking about if there exist a minimal condition to obtain \begin{equation} \mathbb{E}(\log(X))<\infty \end{equation} of course if $X$ is defined to takes values on an interval $[a,b]$ with $a,b>0$ the statement holds, but I think it is very restrictive. An example of what I am asking is: is there exist a $p$ such that $\mathbb{E}(X^p)<\infty$ implies that $log(X)$ has finite expectation? or anyone knows some weak conditions? Any help is welcome, thanks!