We derive asymptotic formulae for the coefficients of bivariate generating functions with algebraic and logarithmic factors.  Logarithms appear when encoding cycles of combinatorial objects, implicitly when objects can be broken into indecomposable parts, and in Póyla enumeration.  Asymptotics are quickly computable and can verify combinatorial properties of sequences and assist in randomly generating objects.  While multiple approaches for algebraic asymptotics have recently emerged, we find that the contour manipulation approach can be extended to generating functions of this type. As an example, we apply the result to the generating function for cyclic partitions of an n-set.