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We apply our previous results on ``saturated descent'' to express a wide range of logarithmic cohomology theories in terms of the infinite root stack. Examples include the log cotangent complex, Rognes' log topological cyclic homology, and Nygaard-complete log prismatic cohomology. As applications, we show that the Nygaard-completion of the site-theoretic log prismatic cohomology coincides with the definition arising from log TC, and we establish a log version of the TC-variant of the Beilinson fiber square of Antieau--Mathew--Morrow--Nikolaus.
Binda et al. (Wed,) studied this question.