Are there any real-valued functions, besides logarithms, for which $f(x*y) = f(x) + f(y)$?

Question

Is there any real-valued function, $f$, which is not a logarithm, such that $∀ x,y$ in $ℝ$ , $f(x*y) = f(x) + f(y)$?

So far, all I can think of is $z$ where $z(x) = 0$ $∀ x$ in $ℝ$

EDIT:

Functions having a domain of $ℝ^+$ or a domain of $ℝ$/{0} are acceptable as well.

What are examples of functions, $f$, from $ℝ$/{0} to $ℝ$ which are not logarithms, such that
$∀ x,y$ in $ℝ$, $f(x*y) = f(x) + f(y)$?

Answer 1

Yes, there are, at least if you assume the axiom of choice. Then there are functions $g\colon\mathbb{R}\longrightarrow\mathbb{R}$ which are not linear but which satisfy Cauchy's functional equation:

$g(x+y)=g(x)+g(y)$.

Now, define

$f\colon(0,+\infty)\longrightarrow\mathbb R$
by
$f(x)=g\bigl(\log(x)\bigr)$.

Answer 2

$f(0\times 0)=f(0)+f(0)$ so $f(0)=0$. Put $x=0$ to get $f(0)=f(0)+f(y)$. Hence $f \equiv 0$. Note that this is the answer when the domain is the whole real line.