We consider the unexploited/exploited logistic equation and study the stability of equilibrium points through Lyapunov functions. Then, we apply first- and second-order optimality conditions for the optimal control of the total biomass yield. Finally, we note that the time-scale logistic equation present in the literature lacks biological significance, and we propose a new version of a dynamic logistic equation, valid on an arbitrary time scale, for which any trajectory of the system, beginning with a positive initial condition, remains nonnegative.