We prove that for no nontrivial ordered abelian group G, the ordered power series field R((G)) over the reals R admits an exponential, i.e., an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a non-surjective logarithm. For an arbitrary ordered field k, no exponential on k((G)) is compatible, that is, induces an exponential on k through the residue map. This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable.

Last update: February 3, 1999