Abstract: Purpose – In many real-life situations ranging from financial to volcanic data, growth is described either by a power law – which is linear in log-log scale or by a quadratic dependence in the log-log scale. The purpose of this paper is to explain this empirical fact. Design/methodology/approach – The authors use natural scale invariance requirements. Findings – In this paper, the authors used natural scale invariance requirement to explain the ubiquity of quadratic log-log dependencies. The authors also explain what to do if quadratic log-log models turn out to be insufficiently accurate. In this case, scale invariance requirements lead to dependencies which in the log-log scale take cubic, 4th order, etc. form. Originality/value – To the best of authors’ knowledge, this is the first theoretical explanation of the empirical quadratic log-log dependence.