  We examine physical aspects for the electric version of a recently proposed
logarithmic electrodynamics, for which the electric field of a point-like
charge is finite at the origin. It is shown that this electrodynamics displays
the vacuum birefringence phenomenon in the presence of external magnetic field.
Afterwards we compute the lowest-order modification to the interaction energy
by means of the gauge-invariant but path-dependent variables formalism. These
are shown to result in a long-range ($1/r^3$-type) correction, in addition to a
linear and another logarithmic correction, to the Coulomb potential.
