Another logarithmic functional equation

Summary. Let f : ]0,￥? \Bbb R 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) f(x + y) - f(x) - f(y) = f(x^{-1} + y^{-1}) ¶and¶f(xy) = f(x) + f(y) f(xy) = f(x) + f(y) ¶are equivalent to each other.