  We define the simplest log-euclidean geometry. This geometry exposes a
difficulty hidden in Hilbert's list of axioms presented in his "Grundlagen der
Geometrie". The list of axioms appears to be incomplete if the foundations of
geometry are to be independent of set theory, as Hilbert intended. In that case
we need to add a missing axiom. Log-euclidean geometry satisfies all axioms but
the missing one, the fifth axiom of congruence and Euclid's axiom of parallels.
This gives an elementary proof (with no need of Riemannian geometry) of the
independence of these axioms from the others.
