  We pursue an investigation of Logarithmic Electrodynamics, for which the
field-energy of a point-like charge is finite, as it happens in the case of the
usual Born-Infeld electrodynamics. We also show that, contrary to the latter,
Logarithmic Electrodynamics exhibits the feature of birefringence. Next, we
analyze the lowest-order modifications for both Logarithmic Electrodynamics and
for its non-commutative version, within the framework of the gauge-invariant
path-dependent variables formalism. The calculation shows a long-range
correction ($1/r^5$- type) to the Coulomb potential for Logarithmic
Electrodynamics. Interestingly enough, for its non-commutative version, the
interaction energy is ultraviolet finite. We highlight the role played by the
new quantum of length in our analysis.
