The derivation on the differential-valued field $\mathbb{ T}_{ \log}$ of logarithmic transseries induces on its value group $\Gamma_{ \log}$ a certain map $\psi$. The structure $(\Gamma_{ \log},\psi)$ is a divisible asymptotic couple. We prove that the theory $T_{ \log} = { \rm Th}(\Gamma_{ \log},\psi)$ admits elimination of quantifiers in a natural first-order language. All models $(\Gamma,\psi)$ of $T_{ \log}$ have an important discrete subset $\Psi:=\psi(\Gamma\setminus\{ 0\})$. We give explicit descriptions of all definable functions on $\Psi$ and prove that $\Psi$ is stably embedded in $\Gamma$.
Categories: math.LO