Logarithmic TC via the infinite root stack and the Beilinson fiber square

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.