Following a suggestion by Mathew, we will explain how to identify some arithmetic invariants of logarithmic schemes. A key ingredient is a form of flat descent for logarithmic invariants that we call "saturated descent”: using this, we compute logarithmic invariants in terms of classical (non-logarithmic) invariants and generalize classical results to logarithmic invariants. As a sample application, we can identify the homotopy-theoretic version of logarithmic prismatic and syntomic cohomology with the site-theoretic version introduced by Koshikawa and Koshikawa-Yao. We will explain how to get from this a log variant of the (graded version of the) Beilinson fiber square of Antieau-Mathew-Morrow-Nikolaus.This is a joint work with F. Binda, T. Lundemo and D. Park.