We present an interpolation method for the specific heat c v (T), when there is a phase transition with a logarithmic singularity in c v at a critical temperature T = T c. The method uses the fact that c v is constrained both by its high temperature series expansion and just above T c by the type of singularity. We test our method on the ferro-and antiferromagnetic Ising models on the two-dimensional square, triangular, honeycomb, and kagome lattices, where we find an excellent agreement with the exact solutions. We then explore the XXZ Heisenberg model, for which no exact results are available.