Circular arrays of log-periodic (LP) antennas are designed and their operational properties are investigated in a sophisticated simulation environment that is based on the recent advances in computational electromagnetics. Due to the complicated structures of the trapezoidal-tooth array elements and the overall array configuration, their analytical treatments are prohibitively difficult. Therefore, the simulation results presented in this paper are essential for their analysis and design. We present the design of a three-element LP array showing broadband characteristics. The directive gain is stabilized in the operation band using optimization by genetic algorithms. We demonstrate that the optimization procedure can also be used to provide beam-steering ability to LP arrays.