  In the probe limit, we investigate the nonlinear electrodynamical effects of
the both exponential form and the logarithmic form on the holographic
paramagnetism-ferromagnetism phase transition in the background of a
Schwarzschild-AdS black hole spacetime. Moreover, by comparing the exponential
form of nonlinear electrodynamics with the logarithmic form of nonlinear
electrodynamics and the Born-Infeld nonlinear electrodynamics which has been
presented in Ref.~\cite{Wu:2016uyj}, we find that the higher nonlinear
electrodynamics correction makes the critical temperature smaller and the
magnetic moment harder form in the case without external field. Furthermore,
the increase of nonlinear parameter b will result in extending the period of
the external magnetic field. Especially, the effect of the exponential form of
nonlinear electrodynamics on the periodicity of hysteresis loop is more
noticeable.
