Logarithmic prismatic cohomology, motivic sheaves, and comparison theorems

We prove that (logarithmic) prismatic and (logarithmic) syntomic cohomologyare representable in the category of logarithmic motives. As an application, weobtain Gysin maps for prismatic and syntomic cohomology, and we explicitlyidentify their cofibers. We also prove a smooth blow-up formula and we computeprismatic and syntomic cohomology of Grassmannians. In the second part of the paper, we develop a descent technique inspired bythe work of Nizio\l~ on log $K$-theory. Using the resulting \emph{saturateddescent}, we prove de Rham and crystalline comparison theorems for logprismatic cohomology, and the existence of Gysin maps for$A_{\inf}$-cohomology.