  We use the techniques of supersymmetric localization to compute the BPS black
hole entropy in N=2 supergravity. We focus on the n_v+1 vector multiplets on
the black hole near horizon background which is AdS_2 x S^2 space. We find the
localizing saddle point of the vector multiplets by solving the localization
equations, and compute the exact one loop partition function on the saddle
point. Furthermore, we propose the appropriate functional integration measure.
Through this measure, the one loop determinant is written in terms of the
radius of the physical metric, which depends on the localizing saddle point
value of the vector multiplets. The result for the one loop determinant is
consistent with the logarithmic corrections to the BPS black hole entropy from
vector multiplets.
