  We propose a class of actions for the spacetime metric that introduce
corrections to the Einstein-Hilbert Lagrangian depending on the logarithm of
some curvature scalars. We show that for some choices of these invariants the
models are ghost free and modify Newtonian gravity below a characteristic
acceleration scale given by a_0 = c\mu, where c is the speed of light and \mu
is a parameter of the model that also determines the late-time Hubble constant:
H_0 \sim \mu. In these models, besides the massless spin two graviton, there is
a scalar excitation of the spacetime metric whose mass depends on the
background curvature. This dependence is such that this scalar, although almost
massless in vacuum, becomes massive and effectively decouples when one gets
close to any source and we recover an acceptable weak field limit at short
distances. There is also a (classical) ``running'' of Newton's constant with
the distance to the sources and gravity is easily enhanced at large distances
by a large ratio. We comment on the possibility of building a model with a
MOND-like Newtonian limit that could explain the rotation curves of galaxies
without introducing Dark Matter using this kind of actions. We also explore
briefly the characteristic gravitational phenomenology that these models imply:
besides a long distance modification of gravity they also predict deviations
from Newton's law at short distances. This short distance scale depends on the
local background curvature of spacetime, and we find that for experiments on
the Earth surface it is of order \sim 0.1mm, while this distance would be
bigger in space where the local curvature is significantly lower.
