An $O_K$-basis for the image of a Lubin-Tate logarithm on $\pi$-regular extensions of $K$

Let $K$ be a finite $p$-adic field with uniformiser $\pi$. In this paper we study the image of the logarithm attached to a Lubin-Tate series $[\pi ](X)$ on the maximal ideal of so-called $\pi$-regular extensions of $K$; for such an extension $L\mathrm{\mid }K$ we compute a basis for the additive group ${\log }_{[\pi ]}(\mathcal{F}({\mathfrak{m}}_L))$ as an $O_K$-module, where $\mathcal{F}({\mathfrak{m}}_L)$ denotes the maximal ideal ${\mathfrak{m}}_L$ equipped with the $O_K$-module structure coming from the formal group associated to $[\pi ](X)$, and determine the minimal valuation of the elements in ${\log }_{[\pi ]}(\mathcal{F}({\mathfrak{m}}_L))$. In the final section of this paper we discuss how some of these results extend to arbitrary finite extensions of $K$ and conclude by determining a basis of the $O_K$-module ${\log }_{[\pi ]}(\mathcal{F}({\mathfrak{m}}_{K_{{\pi }^n}}))$, where $K_{{\pi }^n}$ is the Lubin-Tate extension of level $n\ge 1$.