Coupling of the Legendre polynomials with kernels $|x-y|^\alpha$ and $\ln|x-y|$

Double integrals that represent matrix elements of the power and logarithmic potentials $|x-y|^\alpha$ and $\ln|x-y|$ in the Legendre polynomial basis are found in a closed form. Several proofs are given, which involve different special functions and identities. In particular, a connection of the new formulae and Whipple's hypergeometric summation formula is shown.