A function f with domain $\displaystyle (0,\infty)$ and co-domain $\displaystyle R$ is log-like if

$\displaystyle f(x+y)=f(x)+f(y)$

Show the set of all log-like functions is closed under addition

Attempted Proof

Let functions f and g be log-like

Consider f+g

$\displaystyle f(xy)+g(xy)=f(x)+f(y)+g(x)+g(y)

$

$\displaystyle (f+g)(xy)=(f+g)(x)+(f+g)(y)$

Thus this set is closed

Is this operation correct?