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There exists compelling experimental evidence in numerous systems for logarithmically slow time evolution, yet its full theoretical understanding remains elusive. We here introduce and study a generic transition process in complex systems, based on nonrenewal, aging waiting times. Each state of the system follows a local clock initiated at . The random time between clock ticks follows the waiting time density . Transitions between states occur only at local clock ticks and are hence triggered by the local forward waiting time, rather than by . For power-law forms () we obtain a logarithmic time evolution of the state number ⟨n(t)⟩≃log⁡(t/t0), while for the process becomes normal in the sense that . In the intermediate range we find the power-law growth . Our model provides a universal description for transition dynamics between aging and …