In this paper we prove that the disc is a maximizer of the Schatten
p-norm of the logarithmic potential operator among all domains of a given measure in R², for all 2 ≤ p ≤ ∞. We also show that the equilateral triangle has the largest Schatten p-norm among all triangles of given area. For the logarithmic potential operator on bounded open or triangular domains, we also obtain analogues of the Rayleigh-Faber-Krahn or Pólya inequalities, respectively. The logarithmic potential operator can be related to a nonlocal boundary value problem for the Laplacian, so we obtain isoperimetric inequalities for its eigenvalues as well.