  We present exact black hole solutions endowed with magnetic charge coming
from exponential and logarithmic nonlinear electrodynamics (NLED). Classically,
we analyze the null and timelike geodesics, all of which contain both the bound
and the scattering orbits. Using the effective geometry formalism, we found
that photon can have nontrivial stable (both circular and non-circular) bound
orbits. The noncircular bound orbits for the one-horizon case mostly take the
form of precessed ellipse. For the extremal and three-horizon cases we find
many-world orbits where photon crosses the outer horizon but bounces back
without hitting the true (or second, respectively) horizon, producing the
epicycloid and epitrochoid paths. Semiclassically, we investigate their Hawking
temperature, stability, and phase transition. The nonlinearity enables black
hole stability with smaller radius than its RN counterpart. However, for
very-strong nonlinear regime, the thermodynamic behavior tends to be
Schwarzschild-like.
