Proceedings of the American mathematical society vol:137 issue:5 pages:1663-1676
We study a q-logarithm which was introduced by Euler and give some of its properties. This q-logarithm did not get much attention in the recent literature. We derive basic properties, some of which were already given by Euler in a 1751 paper and 1734 letter to Daniel Bernoulli. The corresponding q-analogue of the dilogarithm is
introduced. The relation to the values at 1 and 2 of a q-analogue of the zeta function is given. We briefly describe some other q-logarithms that have appeared in the recent literature.