Logarithmic Prismatic Cohomology via Logarithmic THH

Inspired by Bhatt-Morrow-Scholze’s work on $\mathrm{T}\mathrm{H}\mathrm{H}$, we define Nygaard-completed log prismatic cohomology based on log topological Hochschild homology via filtrations on log $\mathrm{T}\mathrm{H}\mathrm{H}$ and its variants. Moreover, of independent interest, we describe log $\mathrm{T}\mathrm{H}\mathrm{H}$ for quasiregular semiperfectoids as a $1$-parameter deformation of ordinary, non-logarithmic Hochschild homology.