  In this article, we focus on the error that is committed when computing the
matrix logarithm using the Gauss--Legendre quadrature rules. These formulas can
be interpreted as Pad\'e approximants of a suitable Gauss hypergeometric
function. Empirical observation tells us that the convergence of these
quadratures becomes slow when the matrix is not close to the identity matrix,
thus suggesting the usage of an inverse scaling and squaring approach for
obtaining a matrix with this property. The novelty of this work is the
introduction of error estimates that can be used to select a priori both the
number of Legendre points needed to obtain a given accuracy and the number of
inverse scaling and squaring to be performed. We include some numerical
experiments to show the reliability of the estimates introduced.
