  We investigate Einstein theories of gravity, coupled to a scalar field \vphi
and point-like matter, which are characterized by a scalar field-dependent
matter coupling function e^{H(\vphi)}. We show that under mild constraints on
the form of the potential for the scalar field, there are a broad class of
Einstein-like gravity models -characterized by the asymptotic behavior of H-
which allow for a non-Newtonian weak-field limit with the gravitational
potential behaving for large distances as ln r. The Newtonian term GM/r appears
only as sub-leading. We point out that this behavior is also shared by gravity
models described by f(R) Lagrangians. The relevance of our results for the
building of infrared modified theories of gravity and for modified Newtonian
dynamics is also discussed.
