We consider functions $\omega$ on the unit circle $\bf T$ with a finite number of logarithmic singularities. We study the approximation of $\omega$ by rational functions. We find an asymptotic formula for the distance in the BMO norm between $\omega$ and the set of rational functions of degree $n$ as $n\to\infty$. Our approach relies on the Adamyan-Arov-Krein theorem and on the study of the asymptotic behaviour of singular values of Hankel operators.