An improved particle swarm optimization (PSO) method applied to the design of a new wideband log-periodic antenna (LPA) geometry is introduced. This new PSO variant, called PSO with velocity mutation (PSOvm), induces mutation on the velocities of those particles that cannot improve their position. The proposed LPA consists of wire dipoles with lengths and distances varied according to an exponential rule, which is defined by two specific parameters called length factor and spacing factor. The LPA is optimized for operation in 790-6000 MHz frequency range, in order to cover the most usual wireless services in practice, and also to provide in this range the highest possible forward gain, gain flatness below 2 dB, secondary lobe level below -20 dB with respect to the main lobe peak, and standing wave ratio below 2. To demonstrate its superiority in terms of performance, PSOvm is compared with well-known optimization methods. The comparison is performed by applying all the methods on several test functions and also on the LPA optimization problem defined by the above-mentioned requirements. Furthermore, the radiation characteristics of the PSOvm-based LPA give prominence to the effectiveness of the proposed exponential geometry compared to the traditional Carrel's geometry.