Construction of logarithmic cohomology theories I

We propose a method for constructing cohomology theories of logarithmic schemes with strict normal crossing boundaries by employing techniques from logarithmic motivic homotopy theory over $\mathbb{F}_1$. This method recovers the K-theory of the open complement of a strict normal crossing divisor from the K-theory of schemes as well as logarithmic topological Hochschild homology from the topological Hochschild homology of schemes. In our applications, we establish that the K-theory of non-regular schemes is representable in the logarithmic motivic homotopy category, and we introduce the logarithmic cyclotomic trace for the regular log regular case.