Consider the set of triples $(g_1,g_2,g)\in(\Bbb R^+)^3$ such that $$\log g=(\log g_1)(\log g_2)$$
Is there any geometric or information theoretic meaning behind such triples?
We have $$2\log g=2 (\log g_1)(\log g_2)=(\log(g_1g_2))^2-(\log(g_1))^2-(\log(g_2))^2$$