  In this paper, we take into account the dilaton black hole solutions of
Einstein gravity in the presence of logarithmic and exponential forms of
nonlinear electrodynamics. At first, we consider the cosmological constant and
nonlinear parameter as thermodynamic quantities which can vary. We obtain
thermodynamic quantities of the system such as pressure, temperature and Gibbs
free energy in an extended phase space. We complete the analogy of the
nonlinear dilaton black holes with Van der Waals liquid-gas system. We work in
the canonical ensemble and hence we treat the charge of the black hole as an
external fixed parameter. Moreover, we calculate the critical values of
temperature, volume and pressure and show they depend on dilaton coupling
constant as well as nonlinear parameter. We also investigate the critical
exponents and find that they are universal and independent of the dilaton and
nonlinear parameters, which is an expected result. {Finally, we explore the
phase transition of nonlinear dilaton black holes by studying the Gibbs free
energy of the system. We find that in case of $T>T_c$, we have no phase
transition. When $T=T_c$, the system admits a second order phase transition,
while for $T=T_{\rm f}<T_c$ the system experiences a first order transition.
Interestingly, for $T_{\rm f}<T<T_c$ we observe a \textit{zeroth order} phase
transition in the presence of dilaton field. This novel \textit{zeroth order}
phase transition is occurred due to a finite jump in Gibbs free energy which is
generated by dilaton-electromagnetic coupling constant, $\alpha$, for a certain
range of pressure.}
