  Based on the Sturm-Liouville (SL) eigenvalue problem, we analytically study
several properties of holographic $s$-wave superconductors with exponential
nonlinear electrodynamics in the background of Schwarzschild anti-de Sitter
(AdS) black holes. We assume the probe limit in which the scalar and gauge
fields do not back react on the background metric. We show that for this
system, one can still obtain an analytical relation between the critical
temperature and the charge density. Interestingly enough, we find that
logarithmic nonlinear electrodynamics decreases the critical temperature,
$T_c$, of the holographic superconductors compared to the linear Maxwell field.
This implies that the nonlinear electrodynamics make the condensation harder.
The analytical results obtained in this paper are in good agreement with the
existing numerical results. We also compute the critical exponent near the
critical temperature and find out that it is still $1/2$ which seems to be an
universal value in mean field theory.
