198. The functional equation satisfied by $\log x$

1. It can be shown that there is no solution of the equation (1) which possesses a differential coefficient and is fundamentally distinct from $\log x$. For when we differentiate the functional equation, first with respect to $x$ and then with respect to $y$, we obtain the two equations $$y{f}^{\prime }(xy)=f^{\prime }(x),x{f}^{\prime }(xy)=f^{\prime }(y);$$ and so, eliminating $f^{\prime }(xy)$, $x{f}^{\prime }(x)=y{f}^{\prime }(y)$. But if this is true for every pair of values of $x$ and $y$, then we must have $x{f}^{\prime }(x)=C$, or $f^{\prime }(x)=C/x$, where $C$ is a constant. Hence $$f(x)=\int \frac{C}{x}dx+C^{\prime }=C\log x+C^{\prime },$$ and it is easy to see that $C^{\prime }=0$. Thus there is no solution fundamentally distinct from $\log x$, except the trivial solution $f(x)=0$, obtained by taking $C=0$.

2. Show in the same way that there is no solution of the equation $$f(x)+f(y)=f\left(\frac{x+y}{1–xy}\right)$$ which possesses a differential coefficient and is fundamentally distinct from $\arctan x$.