Another logarithmic functional equation

Summary.

Let $f :\,]0,\infty[\to \Bbb R$ be a real valued function on the set of positive reals. The functional equations¶¶$f(x + y) - f(x) - f(y) = f(x^{-1} + y^{-1})$¶and¶$f(xy) = f(x) + f(y)$¶are equivalent to each other.