Inspired by Bhatt–Morrow–Scholze’s work on THH, we define Nygaard-completed log prismatic cohomology based on log topological Hochschild homology via filtrations on log THH and its variants. Moreover, of independent interest, we describe log THH for quasiregular semiperfectoid pre-log rings as a 1-parameter deformation of ordinary, non-logarithmic Hochschild homology.