Monoid schemes (a.k.a. $\frak M$-schemes) have been introduced by Deitmar as a possible approach to geometry over the field with one element. These build upon monoids as the basic building blocks for algebraic geometry in the place of rings. There's also a variant of Connes--Consani which uses pointed monoids instead of monoids, giving so-called $\frak Mo$-schemes. Connes--Consani's framework is in a sense more general than Deitmar's, as the latter only gives toric schemes.
On the other side, logarithmic schemes (a.k.a. log schemes) are a notion developed with a view to smoothness problems, especially in the theory of moduli, roughly allowing "logarithmic singularities" to be considered as smooth. One defines a log structure on a scheme $X$ to be a sheaf of commutative unpointed monoids $\mathscr{M}$ on $X$ together with a homomorphism of sheaves of monoids $\alpha\colon\mathscr{M}\to\mathscr{O}_X$ such that $\alpha^{-1}(\mathscr{O}^\times_X)\cong\mathscr{O}^\times_X$.
Since monoids, and even monoid schemes (see Ogus's CUP book, Part II, Section 1.2) appear prominently in both theories, it is natural to ask how the two interact with each other. In particular, from the above we see that a log structure on a scheme $X$ may be thought of as a morphism of $\frak M$-schemes $\alpha$ from $(X,\mathscr{M})$ to $(X,\mathscr{O}_X)$ such that $\alpha^{-1}(\mathscr{O}^\times_X)\cong\mathscr{O}^\times_X$.
What is the precise connection/motivation, besides this superficial similarity, between the two theories?
Furthermore, in the case of geometry over the field with one element, it is beneficial to consider not monoids, but pointed monoids, passing from $\frak M$-schemes to $\frak Mo$-schemes.
Is it useful to work with log structures carrying sheaves of pointed commutative monoids instead of sheaves of unpointed commutative monoids?
Edit. Ogus himself mentions $\bf F_1$ in a non-trivial way twice in his book, first on page 192:
and then on page 213:
