  We consider the random-bond Potts model in the large-$Q$ limit and calculate
the excess entropy, $S_{\Gamma}$, of a contour, $\Gamma$, which is given by the
mean number of Fortuin-Kasteleyn clusters which are crossed by $\Gamma$. In two
dimensions $S_{\Gamma}$ is proportional to the length of $\Gamma$, to which -
at the critical point - there are universal logarithmic corrections due to
corners. These are calculated by applying techniques of conformal field theory
and compared with the results of large scale numerical calculations. The
central charge of the model is obtained from the corner contributions to the
excess entropy and independently from the finite-size correction of the
free-energy as: $\lim_{Q \to \infty}c(Q)/\ln Q =0.74(2)$, close to previous
estimates calculated at finite values of $Q$.
