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Here is another simple (symmetrization) trick, which may be better known, though. Let $Y$ be an independent copy of $X$. Note that $x^a\ln x$ is increasing in $x\ge1$, for each real $a\ge0$. So, twice the difference between the left- and right-hand sides of the inequality $(*)$ in question is $$(**)\qquad E(Y-X)(Y^a\ln Y-X^a\ln X)\ge0,$$ since $(Y-X)(Y^a\ln Y-X^a\ln X)\ge0$. Inequality $(**)$, and hence $(*)$, are strict iff the random variable $X$ is non-degenerate; that is, iff $P(X=a)<1$ for all real $a$. We also see that the assumptions $X>e$ and $0<a<1$ can be relaxed to $X\ge1$ and $a\ge0$.