On Heuvers' logarithmic functional equation (Q1411427)

\textit{K. Heuvers} [Aequationes Math. 58, No.~3, 260--264 (1999; Zbl 0939.39016)] has shown that the solutions of the functional equation \[ f(x+y)-f(x)-f(y)=f(x^{-1}+y^{-1}) \] is equivalent to the Cauchy logarithmic functional equation \[ f(xy)=f(x)+f(y) \] when \(f\) is a function from the positive real numbers into the reals. In this paper the author generalizes this result for functions mapping the set of nonzero elements of a commutative field (excluding \(Z_2\)) into any abelian group which has no \(2\)-torsion.