Why does $f(xy) = f(x)+ f(y) \implies f(x) = k \ln x$

Question

My textbook says $f(xy) = f(x)+ f(y) \implies f(x) = k \ln x$

But if I'm not mistaken, $f(x)$ could be a logarithmic function with any base? So why only $\ln x$?

Answer 1

Because $\log_ax=\frac1{\ln a}\ln x$. So, a generic logarithmic function is always a multiple of the natural logarithm.

Note that I am not asserting that these are the only solutions. What I am saying is that the multiples of the natural logarithm contain the logarithms in arbitrary bases.

Answer 2

This is false. If we let $g(x)=f(e^x),$ then we find that $g$ satisfies Cauchy's functional equation, and hence there are infinitely many pathological $g$ which correspond to solutions $f=g(\ln x).$ You must place some continuity or differentiability restrictions on $f.$