Here is another simple (symmetrization) trick, which may be better known, though. Let $Y$ be an independent copy of $X$. Note that $\ln x$ is increasing in $x>0$. So, twice the difference between the left- and right-hand sides of the inequality $(*)$ in question is 
$$(**)\qquad E(Y-X)(\ln Y-\ln X)X^a Y^a\ge0,$$ 
since $(Y-X)(\ln Y-\ln X)X^a Y^a\ge0$. Inequality $(**)$, and hence $(*)$, are strict iff the random variable $X$ is non-degenerate; that is, iff $P(X=b)<1$ for all real $b$. We also see that the assumptions $X>e$ and $0<a<1$ can be relaxed to $X>0$ and $a\in\mathbb{R}$.