  It is known that entropy of black hole gets correction at quantum level.
Universally, these corrections are logarithmic and exponential in nature. We
analyze the impacts of these quantum corrections on thermodynamics of
Born-Infeld BTZ black hole in massive gravity by considering both such kinds of
correction. We do comparative analysis of corrected thermodynamics with their
equilibrium values. Here, we find that the exponential correction yields to the
second point of the first order phase transition. Also, quantum correction
effects significantly on the Helmholtz free energy of larger black holes. We
study the equation of state for the exponential corrected black hole to obtain
a leading order virial expansion.
