On the non-existence of a general Benford's law

Benford's law states that in randomly collected numbers certain digits are more often leading digits than others. More formally: the general law states that the mantissae follow the logarithmic distribution in any base. Benford's law was recognized by many mathematicians so that several possible explanations have been derived, but several questions are still open. Applications are widespread, for example an auditing technique (the so-called digital analysis), which is employed around the world by internal revenue services to detect tax fraud, is based on this phenomenon.In this paper it will be shown that there exists no probability measure that would obey Benford's law for any base, but if the set of possible bases does not exceed a given upper limit, most real-life distributions obey, or can be transformed to obey Benford's law.