The asymptotic couple of the field of logarithmic transseries

The derivation on the differential-valued field TlogTlog of logarithmic transseries induces on its value group ΓlogΓlog a certain map ψ  . The structure (Γlog,ψ)(Γlog,ψ) is a divisible asymptotic couple. We prove that the theory Tlog=Th(Γlog,ψ)Tlog=Th(Γlog,ψ) admits elimination of quantifiers in a natural first-order language. All models (Γ,ψ)(Γ,ψ) of TlogTlog have an important discrete subset Ψ:=ψ(Γ∖{0})Ψ:=ψ(Γ∖{0}). We give explicit descriptions of all definable functions on Ψ and prove that Ψ is stably embedded in Γ.