Another logarithmic functional equation (Q1818259)

The author shows that for real valued functions \(f\) from the set of positive reals \((0, \infty)\) into the set of real numbers \( R\), the classical Cauchy equation \( f(xy)=f(x) + f(y) \) is equivalent to the condition: \( f(x + y) - f(x) - f(y) = f(x^{-1} + y^{-1})\).