  We show how one can obtain rational approximants for $q$-extensions of the
harmonic series and the logarithm (and many other similar quantities) by Pad\'e
approximation using little $q$-Legendre polynomials and we show that properties
of these orthogonal polynomials indeed prove the irrationality, with an upper
bound of the measure of irrationality which is as sharp as the upper bound
given by Bundschuh and V\"a\"an\"anen for the harmonic series and a better
upper bound than the one given by Matala-aho and V\"a\"an\"anen for the
logarithm.
