In this paper, a wideband and low radar cross section (RCS) log-periodic top-hat monopole antenna with a complete ground structure is presented. The study involves an analysis of the influence of the structure on scattering size for the log-periodic antenna with a complete ground structure, particularly when subjected to incident waves grazing from the end-fire direction. Building upon this analysis, the optimal arrangement of the metal patch and its placement at the apex of the antenna is adjusted to achieve a reduction in monostatic RCS. The RCS reduction spans a bandwidth covering from 2 to 13 GHz, with most of the reductions occurring within the entire frequency band, exceeding 5 dB. The maximum RCS reduction is observed at 3.1 GHz, registering at 20.9 dB.