Is the log-function the only function that enables the transformation of a product to a sum:
$$f(x_1\cdot x_2)=f(x_1)+f(x_2)\,?$$
Yes, I can approximate the log function by a Taylor Series, but are there different functions that fulfill this property?
In order to extend the question, is there a (bijective) function (which is also defined for negative values) $f:\mathbb{R}\setminus\{0\} \to \mathbb{R}$ with this property? (This excludes $\log(|x|)$.)
If possible use laymen terms in your answer.
Edit: I never realized how much we can do with prime numbers. @ECL is answering my original question, so I will accept it, but out of curiosity and maybe for the sake of completeness: Can we conclude that there aren't any other functions besides $log$ and $exp$ in $\mathcal{C}^k(D) , D \subseteq \mathbb{R}^+$ and $k\geq0$ with the property?
Best Answer
No you can't find a bijection on $\mathbb{R}$ with this property, the only function is $f\equiv 0$.
Indeed you have $f(0) = f(0\times x) = f(0) + f(x)$ so that $f(x) = 0$. This is true for all $x\in\mathbb{R}$.
If you exclude the $\{0\}$ again you cannot have a bijection since you have that $f$ must be a even function. Indeed $f(x)=\frac{1}{2}f(x^2)=f(-x)$.
However, forgetting about the bijection and looking just at $\mathbb{R}^+$ as a domain, it's true that $\log$ and its multiples are not the only possible functions. Indeed you may find everywhere discontinuous functions with the required property.
For instance you can build the function $f$ as follows. For each prime $p$ you fix $f(p) = k_p$, with $k_p$ any arbitrary number. Then for any $q\in\mathbb{Q}$ you have that $f(p^q)=qf(p)=qk_p$. The value of $f$ is fixed for all the $x$ which can be written as finite product of this terms, i.e. such that there exist a finite family of prime numbers $\{p_i\}_{i=1\dots N}$ and rational coefficients $\{q_i\}_{i=1\dots N}$ and $$x = \prod_{i=1}^N {p_i}^{q_i}\,.$$ You have indeed $f(x) = \sum_{i=1}^N q_i k_{p_i}$. You can show that each of these $x$ has a unique representation in this form, so that $f$ is well defined for them. For the $x$ which are not representable as such a finite product you define $f(x) = 0$. Then $f$ is well defined on $\mathbb{R}^+$ and your property holds.
Edit: To answer your Edit...
All the continuous functions on $\mathbb{R}^+$ which respect the property are in the form $\alpha \log$ for some real $\alpha$. Actually, if $f$ is continuous at $x=1$ then it's in the form $\alpha\log$.
Indeed assume that f is continuous at $1$. First notice that $f(1) = 0$ since $f(x) = f(1\times x) = f(1) + f(x)$. Now you can prove that $f$ is continuous everywhere since $$\lim_{h\to 0} f(x+h) = \lim_{h\to 0}f(x(1+h/x)) = f(x)+\lim_{h\to 0}f(1+h/x) = f(x)+f(1) = f(x)\,.$$ Now you know that for any rational $q$, $f(e^q) = qf(e)$. By continuity you have that for any $r\in\mathbb{R}$ $$f(e^r) = r f(e)\,.$$ So $$f(x) = f(e^{\log x}) = f(e)\log x = \alpha \log x\,,$$ letting $\alpha=f(e)$.
For the sake of completeness, I don't think it is of much sense to speak of the property for a function defined on a generic $D\subset \mathbb{R}^+$. For instance for a open interval $D=(10,11)$, you have no $x_1,x_2\in(10,11)$ such that $x_1\cdot x_2 \in (10,11)$, which means that any function satisfies the property. However, every time that you have a neighbourhood of $1$ in $D$, and $f$ is continuous at $1$, then in each open connected component $f$ must be in the form $x\mapsto k+\alpha \log x$, with $k$ which can possibly vary in different connected components.