  We study the logarithmic corrections to various CFT partition functions in
the context of the AdS$_4$/CFT$_3$ correspondence for theories arising on the
worldvolume of M2-branes. We utilize four-dimensional gauged supergravity and
heat kernel methods and present general expressions for the logarithmic
corrections to the gravitational on-shell action and black hole entropy for a
number of different supergravity backgrounds. We outline several subtle
features of these calculations and contrast them with a similar analysis of
logarithmic corrections performed directly in the eleven-dimensional uplift of
a given four-dimensional supergravity background. We find results consistent
with AdS/CFT provided that the infinite sum over KK modes on the internal space
is regularized in a specific manner. This analysis leads to an explicit
expression for the logarithmic correction to the Bekenstein-Hawking entropy of
large Kerr-Newmann and Reissner-Nordstr\"om black holes in AdS$_4$. Our results
also have important implications for effective field theory coupled to gravity
in AdS$_4$ and for the existence of scale-separated AdS$_4$ vacua in string
theory, which come in the form of new constraints on the field content and mass
spectrum of matter fields.
