  We study some observational signatures of nonlinearities of the
electromagnetic field. First to all we show the vital role played by
nonlinearities in triggering a material behavior of the vacuum with
$(\varepsilon > 0, \mu <0)$, which corresponds to a ferrimagnetic material.
Secondly, the permittivity and susceptibility induced by nonlinearities are
considered in order to obtain the refractive index via the dispersion relation
for logarithmic electrodynamics. Finally, we consider the electromagnetic
radiation produced by a moving charged particle interacting with a medium
characterized by nonlinearities of the electromagnetic field. To this end we
consider logarithmic electrodynamics. The result shows that the radiation is
driven by the medium through which the particle travels like the one that
happens in the Cherenkov effect.
