  We give a new and conceptually simple approach to obtain the first law of
black hole thermodynamics from a basic thermodynamical property that entropy
(S) for any stationary black hole is a state function implying that dS must be
an exact differential. Using this property we obtain some conditions which are
analogous to Maxwell's relations in ordinary thermodynamics. From these
conditions we are able to explicitly calculate the semiclassical
Bekenstein-Hawking entropy, considering the most general metric represented by
the Kerr-Newman spacetime. We extend our method to find the corrected entropy
of stationary black holes in (3+1) dimensions. For that we first calculate the
corrected Hawking temperature considering both scalar particle and fermion
tunneling beyond the semiclassical approximation. Using this corrected Hawking
temperature we compute the corrected entropy, based on properties of exact
differentials. The connection of the coefficient of the leading (logarithmic)
correction with the trace anomaly of the stress tensor is established . We
explicitly calculate this coefficient for stationary black holes with various
metrics, emphasising the role of Komar integrals.
