  We consider modifications of General Relativity obtained by adding the
logarithm of some curvature invariants to the Einstein-Hilbert action. These
non-linear actions can explain the late-time acceleration of the universe
giving an expansion history that differs from that of a pure cosmological
constant. We show that they also modify the Newtonian potential below a fixed
acceleration scale given by the late-time Hubble constant times the speed of
light. This is exactly what is required in MOND, a phenomenological
modification of the Newtonian potential that is capable of explaining galactic
rotation curves without the need to introduce dark matter. We show that this
kind of modification also predicts short distance deviations of Newton's law at
the sub-mm scale and an anomalous shift in the precession of the Moon's orbit
around the Earth, both effects of a size that is less than an order of
magnitude below current bounds.
