  In this paper, we consider Einstein gravity in the presence of a class of
nonlinear electrodynamics, called power Maxwell invariant (PMI). We take into
account $(2+1)$-dimensional spacetime in Einstein-PMI gravity and obtain its
black hole solutions. Then, we regard pure $F(R)$ gravity as well as
$F(R)$-conformally invariant Maxwell theory to obtain exact solutions of the
field equations with black hole interpretation. Finally, we investigate the
conserved and thermodynamic quantities and discuss about the first law of
thermodynamics for the mentioned gravitational models.
