We study the global dynamics of large amplitude classical solutions to a system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, subject to time-dependent boundary conditions. The model is supplemented with $H^2$ initial data and unmatched boundary conditions at the endpoints of a one-dimensional interval. Under suitable assumptions on the boundary data, it is shown that classical solutions exist globally in time. Time asymptotically, the differences between the solutions and their corresponding boundary data converge to zero, as time goes to infinity. No smallness restrictions on the magnitude of the initial perturbations is imposed. Numerical simulations are carried out to explore some topics that are not covered by the analytical results.