Given an ordered field $K$, we compute the natural valuation and skeleton of the ordered multiplicative group $(K^{>0},\cdot ,1,<)$ in terms of those of the ordered additive group $(K,+,0,<)$. We use this computation to provide necessary and sufficient conditions on the value group $v(K)$ and residue field $\ovl{K}$, for the $\Lio$-equivalence of the above mentioned groups. We then apply the results to exponential fields, and describe $v(K)$ in that case. Finally, if $K$ is countable or a power series field, we derive necessary and sufficient conditions on $v(K)$ and $\ovl{K}$ for $K$ to be exponential. In the countable case, we get a structure theorem for $v(K)$.}

Last update: February 6, 1999