  We construct a new class of charged rotating black brane solutions in the
presence of logarithmic nonlinear electrodynamics with complete set of the
rotation parameters in arbitrary dimensions. The topology of the horizon of
these rotating black branes are flat, while, due to the presence of the dilaton
field the asymptotic behaviour of them are neither flat nor (anti)-de Sitter
[(A)dS]. We investigate the physical properties of the solutions. The mass and
angular momentum of the spacetime are obtained by using the counterterm method
inspired by AdS/CFT correspondence. We derive temperature, electric potential
and entropy associated with the horizon and check the validity of the first law
of thermodynamics on the black brane horizon. We study thermal stability of the
solutions in both canonical and grand canonical ensemble and disclose the
effects of the rotation parameter, nonlinearity of electrodynamics and dilaton
field on the thermal stability conditions. We find the solutions are thermally
stable for $\alpha<1$, while for $\alpha>1$ the solutions may encounter an
unstable phase where $\alpha$ is dilaton-electromagnetic coupling constant.
