The source text

Kato, Kazuya
Logarithmic structures of Fontaine-Illusie. Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 191–224, Johns Hopkins Univ. Press, Baltimore, MD, 1989.

available here

http://www.math.brown.edu/~abrmovic/LOGGEOM/Kato-log.pdf.

Example (3.7) (1)
the bottom line is

"... essentially, the theory of toroidal embeddings is nothing but the theory of schemes with smooth fine log. str.'s over a field (with respect to the trivial log. str. on the base field)."