  We present a numerical method for computing the logarithmic capacity of
compact subsets of $\mathbb{C}$, which are bounded by Jordan curves and have
finitely connected complement. The subsets may have several components and need
not have any special symmetry. The method relies on the conformal map onto
lemniscatic domains and, computationally, on the solution of a boundary
integral equation with the Neumann kernel. Our numerical examples indicate that
the method is fast and accurate. We apply it to give an estimate of the
logarithmic capacity of the Cantor middle third set and generalizations of it.
