  We explore a well-known integral representation of the logarithmic function,
and demonstrate its usefulness in obtaining compact, easily-computable exact
formulas for quantities that involve expectations and higher moments of the
logarithm of a positive random variable (or the logarithm of a sum of positive
random variables). The integral representation of the logarithm is proved
useful in a variety of information-theoretic applications, including universal
lossless data compression, entropy and differential entropy evaluations, and
the calculation of the ergodic capacity of the single-input, multiple-output
(SIMO) Gaussian channel with random parameters (known to both transmitter and
receiver). This integral representation and its variants are anticipated to
serve as a useful tool in additional applications, as a rigorous alternative to
the popular (but non-rigorous) replica method (at least in some situations).
