Logarithmic prismatic cohomology II

We continue to study the logarithmic prismatic cohomology defined by the first author, and complete the proof of the de Rham comparison and étale comparison generalizing those of Bhatt and Scholze. We prove these comparisons for a derived version of logarithmic prismatic cohomology, and, along the way, we construct a suitable Nygaard filtration and explain a relation between $F$-crystals and ${\mathbb{Z}}_p$-local systems in the logarithmic setting.