  The theory of nonlinear electrodynamics has got a lot of attentions in recent
years. It was shown that Born-Infeld nonlinear electrodynamics is not the only
modification of the linear Maxwell's field which keeps the electric field of a
charged point particle finite at the origin, and other type of nonlinear
Lagrangian such as exponential and logarithmic nonlinear electrodynamics can
play the same role. In this paper, we generalize the study on the exponential
nonlinear electrodynamics by adding a scalar dilaton field to the action. By
suitably choosing the coupling of the matter field to the dilaton field, we
vary the action and obtain the corresponding field equations. Then, by making a
proper ansatz, we construct a new class of charged dilaton black hole solutions
coupled to the exponential nonlinear electrodynamics field in the presence of
two Liouville-type potentials for the dilaton field. Due to the presence of the
dilaton field, the asymptotic behavior of these solutions are neither flat nor
(A)dS. In the limiting case where the nonlinear parameter $\beta^2$ goes to
infinity, our solution reduces to the Einstein-Maxwell dilaton black holes. We
obtain the mass, temperature, entropy and electric potential of these
solutions. We also study the behaviour of the electric field as well as the
electric potential of these black holes near the origin. We find that the
electric field has a finite value \textit{near} the origin, which is the same
as the electric field of Born-Infeld nonlinear electrodynamics, but it can
diverge exactly at $r=0$ depending on the model parameters. We also investigate
the effects of the dilaton field on the behaviour of the electric field and
electric potential. Finally, we check the validity of the first law of black
hole thermodynamics on the black hole horizon.
