The wrong direction of Jensen's inequality is algorithmically right

Let $\mathcal{A}$ be an algorithm with expected running time $e^X$, conditioned on the value of some random variable $X$. We construct an algorithm ${\mathcal{A}}^{\mathcal{\prime }}$ with expected running time $O(e^{E[X]})$, that fully executes $\mathcal{A}$. In particular, an algorithm whose running time is a random variable $T$ can be converted to one with expected running time $O(e^{E[\ln T]})$, which is never worse than $O(E[T])$. No information about the distribution of $X$ is required for the construction of ${\mathcal{A}}^{\mathrm{\prime }}$.