A compact and low-profile log-periodic meandered dipole array (LPMDA) antenna with an artificial magnetic conductor (AMC) is proposed. For compactness, a meander line configuration is implemented with dipole elements and optimized using a genetic algorithm (GA) to realize the LPMDA antenna. As a result, a size reduction of approximately 30% is achieved as compared to a conventional log-periodic dipole array antenna. To enhance the gain characteristics, the AMC ground plane configuration is realized with 9 × 9 unit cells for the LPMDA antenna. Two prototypes of the proposed LPMDA antennas with and without an AMC are fabricated and measured to verify its performance. The measured −10 dB reflection ratio bandwidths are 2.56 : 1 (0.85–2.18 GHz) and 2.34 : 1 (0.92–2.16 GHz) for the proposed LPMDA antennas with and without the AMC, respectively. The gain at the main beam direction within the operating frequency bandwidth is significantly improved from 3.94–7.17 dBi to 7.86–10.01 dBi by applying the AMC.