  Probability measures with either finite Monge-Amp\`ere energy or finite
entropy have played a central role in recent developments in K\"ahler geometry.
In this note we make a systematic study of quasi-plurisubharmonic potentials
whose Monge-Amp\`ere measures have finite entropy. We show that these
potentials belong to the finite energy class ${\mathcal E}^{\frac{n}{n-1}}$,
where $n$ denotes the complex dimension, and provide examples showing that this
critical exponent is sharp. Our proof relies on refined Moser-Trudinger
inequalities for quasi-plurisubharmonic functions.
