  We analytically study the properties of a $(2+1)$-dimensional $s$-wave
holographic superconductor in the presence of exponential nonlinear
electrodynamics. We consider the case in which the scalar and gauge fields back
react on the background metric. Employing the analytical Sturm-Liouville
method, we find that in the black hole background, the nonlinear
electrodynamics correction will affect the properties of the holographic
superconductors. We find that with increasing both backreaction and nonlinear
parameters, the scalar hair condensation on the boundary will develop more
difficult. We obtain the relation connecting the critical temperature with the
charge density. Our analytical results support that, even in the presence of
the nonlinear electrodynamics and backreaction, the phase transition for the
holographic superconductor still belongs to the second order and the critical
exponent of the system always takes the mean-field value $1/2$.
