Bounds for the logarithmic function are studied. In particular, we
establish bounds with rational functions as approximants. The study
leads into the fascinating areas of Padé approximations ([2], [6]), continued
fractions ([7], [11]) and orthogonal polynomials ([14], [4]) as
well as the somewhat frightening jungle of special functions and associated
identities ([5], [9]). Originally, the results aimed at establishing
certain inequalities for Shannon entropy but are here discussed in their
own right (the applications to entropy inequalities will be published
elsewhere).
The exposition is informal, a kind of essay, with only occasional
indications of proofs. The reader may take it as an invitation to further
studies. Enough details are provided to enable the reader to verify all
statements. To the expert in the fields pointed to there is little or
nothing new.