Benford law and lognormal distributions

Benford’s law is nowadays extremely popular (see e.g. http://en.wikipedia.org/…). It is usually claimed that, for a given set data set, changing units does not affect the distribution of the first digit. Thus, it should be related to scale invariant distributions. Heuristically, scale (or unit) invariance means that the density of the measure  (or probability function) should be proportional to . Thus, because densities integrate to 1, the proportionality coefficient has to be , and therefore, should satisfy the following functional equation, , for all [...]