  We prove that for no nontrivial ordered abelian group G, the ordered power
series field R((G)) 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.
