A constructive tool of nonlinear control system design, the method of control Lyapunov functions (CLFs), has found numerous applications in stabilization problems for continuous-time, discrete-time, and hybrid systems. In this paper, we address the fundamental question: Given a CLF, corresponding to a continuous-time controller with some predefined (e.g., exponential) convergence rate, can the same convergence rate be provided by an event-triggered controller? Under certain assumptions, we give an affirmative answer to this question and show that the corresponding event-based controllers provide positive dwell times between consecutive events. Furthermore, we prove the existence of self-triggered and periodic event-triggered controllers, providing stabilization with a known convergence rate.