  We disclose the effects of the logarithmic nonlinear electrodynamics on the
holographic conductivity of Lifshitz dilaton black holes/branes. We analyze
thermodynamics of these solutions as a necessary requirement for applying
gauge/gravity duality, by calculating conserved and thermodynamic quantities
such as the temperature, entropy, electric potential and mass of the black
holes/branes. We calculate the holographic conductivity for a
$(2+1)$-dimensional brane boundary and study its behavior in terms of the
frequency per temperature. Interestingly enough, we find out that, in contrast
to the Lifshitz-Maxwell-dilaton black branes which has conductivity for all
$z$, here in the presence of nonlinear gauge field, the holographic
conductivity do exist provided $z\leq3$ and vanishes for $z>3$. It is shown
that independent of the nonlinear parameter $\beta$, the real part of the
conductivity is the same for a specific value of frequency per temperature in
both AdS and Lifshitz cases. Besides, the behavior of real part of conductivity
for large frequencies has a positive slope with respect to large frequencies
for a system with Lifshitz symmetry whereas it tends to a constant for a system
with AdS symmetry. This behavior may be interpreted as existence of an
additional charge carrier rather than the AdS case, and is due to the presence
of the scalar dilaton field in model. Similar behavior for optical conductivity
of single-layer graphene induced by mild oxygen plasma exposure has been
reported.
