On the relationship between logarithmic TAQ and logarithmic THH

We provide a new description of logarithmic topological Andre-Quillen homology in terms of the indecomposables of an augmented ring spectrum. The new description allows us to interpret logarithmic TAQ as an abstract cotangent complex, and leads to an etale descent formula for logarithmic topological Hochschild homology. The latter is analogous to results of Weibel-Geller for Hochschild homology of discrete rings, and of McCarthy-Minasian and Mathew for topological Hochschild homology. We also summarize and clarify analogous results relating notions of formal etaleness defined in terms of ordinary THH and TAQ.