  A family of singularly perturbed q-difference-differential equations under
the action of a small complex perturbation parameter is studied. The action of
the formal monodromy around the origin is present in the equation, which
suggests the construction of holomorphic solutions holding logarithmic terms in
both, the formal and the analytic level. We provide both solutions and describe
the asymptotic behavior relating them by means of $q-$gevrey asymptotic
expansions of some positive order, with respect to the perturbation parameter.
  On the way, the development of a space product of Banach spaces in the Borel
plane is needed to provide a fixed point for a coupled system of equations.
