This is in reference to to this post, where B. Goddard says that all functions of the type $a \ln x -\frac 1x$ satisfy the functional equation $$ f(xy) = f(x) + f(y) + \frac{x+y-1}{xy} $$ And I was looking for a concrete proof of the above. I went about and tried to solve it as follows:
In accordance with B. Goddard's statement that $f(x)$ must have a "$\ln x$" component, I wrote $f(x)=a\ln x + g(x)$ and on substitution, what remained was $$g(xy)+\frac{1}{xy}= g(x)+\frac 1x +g(y)+\frac 1y$$ From here, it is easy to guess the $g(x)=-\frac 1x$ solution.
But I felt that this was more guesswork than concrete math. So if you could help me with the proof, I'd be very happy.
Thanks in advance.
Guessing that $f(x) = - \frac{1}{x} $ is a solution is quite "obvious".
A natural next step is to look at $ h(x) = f(x) + \frac{ 1}{x}$, from which we have $h(xy) = h(x) + h(y)$, which is a Cauchy functional, with solutions $h(x) = a \ln x $.
I'm guessing that this is what Goddard means by "Because of the product to sum property", where we hope to reduce it to a form like $h(xy) = h(x) + h(y)$.