I would like to understand if the logarithmic blow-up of a log scheme can be performed étale locally:
Specifically, suppose that $X^{\dagger}=(X,\mathcal{M}_{X})$ is a fine, saturated log scheme and $U^{\dagger}=(U,\mathcal{M}_{U})$ is a strict étale open, namely, there is a morphism $f:U^{\dagger}\rightarrow X^{\dagger}$ of log schemes such that $f:U\rightarrow X$ is étale in the usual sense and $f^{\flat}:\mathcal{M}_{X}\rightarrow f_{*}\mathcal{M}_{U}$ is an isomorphism.
Suppose further that $g:U^{\dagger}\rightarrow Z^{\dagger}=\text{Spec}(\mathbb{Z}[P])$ is a chart given by a fine, saturated monoid $P$ (so a local chart for $X^{\dagger}$) and $I\subseteq P$ is a finitely generated ideal. Then the logarithmic blow-up of $U^{\dagger}$ with respect to $I$ is given by the pullback of the logarithmic blow-up $Z_{I}^{\dagger}\rightarrow Z^{\dagger}$ along $g$. (Specifically, we blow up $Z$ at centre $\langle I \rangle \leq \mathbb{Z}[P]$; and saturate the natural fine log structure). We denote this log scheme by $U_{I}^{\dagger}$.
I would like to know if there exists a log scheme $\tilde{X}^{\dagger}\rightarrow X^{\dagger}$ such that $U_{I}^{\dagger}\rightarrow U^{\dagger}$ is the pullback of this morphism along the étale open $f:U\rightarrow X$ and such that the induced morphism $U_{I}^{\dagger}\rightarrow \tilde{X}^{\dagger}$ is étale.
I understand that, usually, when one works with log schemes, one works in the (small) étale site, however, it is not clear to me why the log scheme $\tilde{X}^{\dagger}$ should exist a priori. Certainly, if one takes an étale open $U$ of a usual scheme $X$ and takes a blow up $\text{Bl}_{I}U$ of $U$, then in general there does not exist a scheme $\tilde{X}$ that is the 'blow-up of $X$ at $I$'.
