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!
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