  There exists compelling experimental evidence in numerous systems for
logarithmically slow time evolution, yet its theoretical understanding remains
elusive. We here introduce and study a generic transition process in complex
systems, based on non-renewal, aging waiting times. Each state n of the system
follows a local clock initiated at t=0. The random time \tau between clock
ticks follows the waiting time density \psi(\tau). Transitions between states
occur only at local clock ticks and are hence triggered by the local forward
waiting time, rather than by \psi(\tau). For power-law forms \psi(\tau) ~
\tau^{-1-\alpha} (0<\alpha<1) we obtain a logarithmic time evolution of the
state number <n(t)> ~ log(t/t_0), while for \alpha>2 the process becomes normal
in the sense that <n(t)> ~ t. In the intermediate range 1<\alpha<2 we find the
power-law growth <n(t)> ~ t^{\alpha-1}. Our model provides a universal
description for transition dynamics between aging and non-aging states.
