Let $M_{g, n}$ be the moduli space of curves of genus $g$ with $n$ marked points. Let $M_{g, \vec{n}}$ be the moduli space of marked curves with a choice of a (possibly zero) tangent vector at each marked point. There is a map $i:M_{g, n}\to M_{g, \vec{n}}$ (the zero section), and there is a log structure on $M_{g, \vec{n}}$ associated to the normal crossings divisor which is the complement to the locus where all specified tangent vectors are nonzero. Let $M_{g, n}^{log}$ be the pullback of this log structure on $M_{g, \vec{n}}$ to $M_{g, n}$ along $i.$

**Question:** what is $M_{g, n}^{log}$ called? Who has studied it? At one point I heard the term "log moduli space of curves with log structure" floating around, though I've never seen it explicitly defined. Is my $M_{g, n}^{log}$ the (restriction from $\bar{M}_{g,n}$ to $M_{g,n}$) of this log moduli space?