In the terahertz frequency range, slow-wave structures (SWSs) in vacuum electronic devices typically exhibit strong dispersion characteristics, which tend to produce low group velocity regions near the band-edge frequencies, posing a risk of oscillation. To address this issue, a novel planar modified angular log-periodic (MALP) slot-line slow wave structure (MSL-SWS) is proposed in this paper. This SWS consists of a central metallic sheet etched with grooves along a modified angular log-periodic trajectory, sandwiched between two metallic enclosures. This configuration effectively mitigates the charge accumulation problems commonly observed in planar microstrip structures. Each unit of the proposed aperiodic structure possesses distinct dispersion characteristics, and the continuous variation in phase velocity across the cells suppresses potential oscillation. Simulation results demonstrate that the structure exhibits favorable transmission bandwidth and achieves stable amplification at 0.22 THz, with a saturated output power of 141 W and a gain of 24.9 dB. These results clearly verify the great potential of the proposed SWS for terahertz vacuum amplifier applications.