  We investigate the heat kernel method for one-loop effective action following
the Seeley-DeWitt expansion technique of heat kernel with Seeley-DeWitt
coefficients. We also review a general approach of computing the Seeley-DeWitt
coefficients in terms of background or geometric invariants. We, then consider
the Einstein-Maxwell theory embedded in minimal $\mathcal{N}=2$ supergravity in
four dimensions and compute the first three Seeley-DeWitt coefficients of the
kinetic operator of the bosonic and the fermionic fields in an arbitrary
background field configuration. We find the applications of these results in
the computation of logarithmic corrections to Bekenstein-Hawking entropy of the
extremal Kerr-Newman, Kerr and Reissner-Nordstrom black holes in minimal
$\mathcal{N}=2$ Einstein-Maxwell supergravity theory following the quantum
entropy function formalism.
