  We study one-loop covariant effective action of \say{non-minimally coupled}
$\mathcal{N}=1$, $d=4$ Einstein-Maxwell supergravity theory by heat kernel
tool. By fluctuating the fields around the classical background, we study the
functional determinant of Laplacian differential operator following
Seeley-DeWitt technique of heat kernel expansion in proper time. We then
compute the Seeley-DeWitt coefficients obtained through the expansion. A
particular Seeley-DeWitt coefficient is used for determining the logarithmic
correction to Bekenstein-Hawking entropy of extremal black holes using quantum
entropy function formalism. We thus determine the logarithmic correction to the
entropy of Kerr-Newman, Kerr and Reissner-Nordstr\"{o}m black holes in
{\say{non-minimally coupled}} $\mathcal{N}=1$, $d=4$ Einstein-Maxwell
supergravity theory.
