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As a first step to understand the theory of the structure $\mathbb{T}_{\log}$of logarithmic transseries as an ordered valued logarithmic field, we focus onthe map $\chi$ induced by the logarithm of $\mathbb{T}_{\log}$ in its valuegroup $\Gamma_{\log}$ and study the theory of the precontraction group$(\Gamma_{\log},\chi)$. Particularly, we show that this theory is modelcomplete and complete, and we characterize all definable subsets of thediscrete set $\chi(\Gamma_{\log})$.

The precontraction group of the field of logarithmic transseries $\mathbb{T}_{\log}$