  In this paper, we find the solutions of quartic quasitopological black holes
and branes coupled to logarithmic and exponential forms of nonlinear
electrodynamics. These solutions have an essential singularity at $r=0$.
Depending on the value of charge parameter $q$, we have an extreme black
hole/brane, a black hole/brane with two horizons or a naked singularity. For
small values of parameter $q$, the solutions lead to a black hole/brane with
two horizons. The values of the horizons are independent of the values of
quasitopological parameters and depend only on the values of $q$, dimensions
$n$, nonlinear parameter $\beta$ and mass parameter. Also, the solutions are
not thermally stable for dS and flat spacetimes. However, AdS solutions are
stable for $r_{+}>{r_{+}}_{\rm{ext}}$ which the temperature is zero for
$r_{+}={r_{+}}_{\rm{ext}}$. The value of ${r_{+}}_{\rm{ext}}$ also depends on
the values of parameters $q$, $\beta$, $n$ and $m$. As the value of
${r_{+}}_{\rm{ext}}$ decreases, the region of stability becomes larger. We also
use HPEM metric to probe GTD formalism for our solutions. This metric is
successful to predict the divergences of the scalar curvature exactly at the
phase transition points. For large values of parameter $\Xi$, the black
hole/brane has a transition to a stable state and stays stable.
