Different functional equations than Cauchy type? $2f(xy)=f(x)+f(y)$

Question

Since we have solution for Cauchy functional equation, $$f(xy)= f(x)+f(y)$$ which is $C\log(x)$. However, if we have $$2f(xy)=f(x)+f(y)$$ type of functional equation, I found that its solutions are any arbitrary constant $C$, but failed to prove.

How do I prove the above claim?

Answer 1

For $y = 1$ we obtain $f(x) = f(1)$ so $f$ is constant.