Best finite approximations of Benford's Law

For arbitrary Borel probability measures with compact support on the real line, characterizations are established of the best finitely supported approximations, relative to three familiar probability metrics (Levy, Kantorovich, and Kolmogorov), given any number of atoms, and allowing for additional constraints regarding weights or positions of atoms. As an application, best (constrained or unconstrained) approximations are identified for Benford's Law (logarithmic distribution of significands) and other familiar distributions. The results complement and extend known facts in the literature; they also provide new rigorous benchmarks against which to evaluate empirical observations regarding Benford's Law.