  We examine the higher dimensional action in which gravity is coupled to the
exponential nonlinear electrodynamic and a scalar dilaton field. We construct a
new class of $n$-dimensional static and spherically symmetric black hole
solutions of this theory in the presence of the dilaton potential with two
Liouville-type terms. In the presence of two Liouville-type dilaton potential,
the asymptotic behavior of the obtained black holes are neither flat nor (A)dS.
Due to the nonlinear nature of electrodynamic field, the electric field has
finite value near the origin where $r\rightarrow0$ and goes to zero as
$r\rightarrow\infty$. Interestingly enough, we find that in the absence of the
dilaton field, the electric field has a finite value at $r=0$, while as soon as
the dilaton field is taken into account, the electric field diverges as
$r\rightarrow 0$. This implies that the presence of the dilaton field changes
the behaviour of the electric field near the origin. In the limiting case where
the nonlinear parameter $\beta$ goes to infinity, our solutions reduce to
dilaton black holes of Einstein-Maxwell-dilaton gravity in higher dimensions.
We compute the conserved and thermodynamic quantities of the solutions and show
that these quantities satisfy the first law of black holes thermodynamics on
the horizon.
