  Recently, different numerical studies of coarsening in disordered systems
have shown the existence of a crossover from an initial, transient, power-law
domain growth to a slower, presumably logarithmic, growth. However, due to the
very slow dynamics and the long lasting transient regime, one is usually not
able to fully enter the asymptotic regime when investigating the relaxation of
these systems toward equilibrium. We here study two simple driven systems, the
one-dimensional $ABC$ model and a related domain model with simplified
dynamics, that are known to exhibit anomalous slow relaxation where the
asymptotic logarithmic growth regime is readily accessible. Studying two-times
correlation and response functions, we focus on aging processes and dynamical
scaling during logarithmic growth. Using the time-dependent growth length as
the scaling variable, a simple aging picture emerges that is expected to also
prevail in the asymptotic regime of disordered ferromagnets and spin glasses.
