A universal Cauchy functional equation over the positive reals

The functional equation a f ( x y ) + b f ( x ) f ( y ) + c f ( x + y ) + d ( f ( x ) + f ( y ) ) = 0 whose shape contains all the four well-known forms of Cauchyʹs functional equation is solved for solutions which are functions having the positive reals as their domain. This complements an earlier work of Dhombres in 1988 where the same functional equation was solved for solutions whose domains contain zero, which leaves out the logarithmic function. Here not only the logarithmic function is recovered but the analysis is entirely different and is based on solving appropriate difference equations.