Logarithmic Geometry was invented (or discovered) in the 1980's, with crucial ideas contributed by Deligne, Faltings, Fontaine, Illusie, and especially Kato. Its roots go back much further, at least to the work by Mumford and others on toroidal embeddings, as well as the theory of semistable reduction. The goal of all these theories is to understand, in a systematic way, the related phenomena of compactification and degeneration in algebraic and arithmetic geometry, with applications to number theory.
I have been working on a manuscript which is intended to serve as a basic introduction to the main ideas and techniques of log geometry for over a decade now. Because of the long scope of the project and the lack of a systematic treatment of these ideas and techniques in the published literature, I have freely shared, upon request, preliminary (and in many ways unsatisfactory) versions of the manuscript. As I warned recipients, these early versions were incomplete and included plans or sketches for proofs and statements that in some cases did not pan out. The most recent revision freely available version was posted in December of 2017; it benefitted from many corrections and suggestions due to colleagues, especially Ofer Gabber.
I am happy to report that, after one more further proofreading by me and copyediting by professionals, the manuscript has now been published by Cambridge University Press. A hardback copy should be available at a list price of $89.99. The electronic version, which costs somewhat less and which, according to my understanding, should be licensed to most academic institutions under the Cambridge Core agreement..
My contract with Cambridge does not permit me to post any version that makes use of their copyediting services. Since the proofreading and copyediting revealed many defects in the version of December 2017, including some miselading mathematical statements, I have decided to remove it from public view. I ask those of you who already have it to replace it with the new version, or if that is not possible, to be very wary in the course of its use. The published version still contains omissions and, I am virtually certain, errors. Please communicate to me any of these that you find. I will do my best to correct these as best I can on this web site; see the link to Errata pages.