functional equations - If $f(xy) = f(x) + f(y)$, show that $f(.)$ can only be a logarithmic function.

As the question states, show that the property exhibited can only be satisfied by a logarithmic function i.e no other family of functions can satisfy the above property.

Answer

Continuity is necessary.

If $F(x+y)=F(x)+F(y)$, for all $x,y$ and $F$ discontinuous (such $F$ exist due to the Axiom of Choice, and in particular, the fact that $\mathbb R$ over $\mathbb Q$ possesses a Hamel basis) and
$f(x)=F(\log x)$, then $f(xy)=f(x)+f(y)$, and $f$ is not logarithmic!