Let P be a homogeneous polynomial in N+1 complex variables of degree d. The logarithmic Mahler Measure of P (denoted by m(P) ) is the integral of log|P| over the sphere in C^{N+1} with respect to the usual Hermitian metric and measure on the sphere.  Now let X be a smooth variety embedded in CP^N by a high power of an ample line bundle and let $\Delta$ denote a generalized discriminant of X wrt the given embedding , then $\Delta$ is an irreducible homogeneous polynomial in the appropriate space of variables.  In this talk I will discuss work in progress whose aim is to find an asymptotic expansion of m(\Delta) in terms of elementary functions of the degree of the embedding.