On the q-factorization of power series

Any power series with unit constant term can be factored into an infinite product of the form ∏n≥1(1-qn)-an. We give direct formulas for the exponents an in terms of the coefficients of the power series, and vice versa, as sums over partitions. As examples, we prove identities for certain partition enumeration functions. Finally, we note q-analogues of our enumeration formulas.