  We present exact black hole solutions endowed with magnetic charge coming
from exponential and logarithmic nonlinear electrodynamics (NLED). 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-ciruclar) bound orbits. The
noncircular bound orbits for the one-horizon case mostly take the form of
precessed ellipse. In the case of extremal (two-horizons) and three-horizon
cases we find conditions where the photon's bound orbit crosses the outer
horizon but bounces back without hitting the true (or second, respectively)
horizon, producing the epicycloid and epitrochoid paths, repsectively. The
validity of such horizon-crossing orbits has been evaluated using the
Eddington-Finkelstein transformation, and it shows that there are indeed
possible. Semiclassically, we investigate their Hawking temperature, stability,
and phase transition. It is shown that for very strong nonlinear parameter, the
thermodynamic behavior tends to be Schwarzschild-like. On the other hand, the
nonlinearity of the matter enables the existence of stable black holes to have
smaller radius than its RN counterpart.
