  We solve a Pad\'e-type problem of approximating three specific functions
simultaneously by $q$-analogues of polylogarithms, respectively by powers of
the logarithm. This problem is intimately related to recent results of the
authors and Wadim Zudilin ["S\'eries hyperg\'eom\'etriques basiques, fonction
$q$-z\^eta et s\'eries d'Eisenstein", J. Inst. Math. Jussieu (to appear)] on
the dimension of the vector space generated by $q$-analogues of values of the
Riemann zeta function at integers. We also show that our result can be
considered as a $q$-analogue of a result of St\'ephane Fischler and the second
author [J. Math. Pures Appl. {\bf 82} (2003), 1369-1394].
