My son was working the other day with exercises such as: Find all the mappings $f:\mathbb{N}\rightarrow\mathbb{Z}$ verifying
$$\forall m,n \in \mathbb{N}, f(m+n)=f(n)+f(m).$$
As another example: Find all the mappings $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$\forall x,y \in \mathbb{R}, f(x)f(y)=f(xy)+x+y.$$
I know that there several equivalent ways to introduce the natural logarithmic function but I was wondering how one can define it properly and consicely by means of the functional relation
$$f(xy)=f(x)+f(y).\qquad (1)$$
That is starting from this relation and not any prior knowledge of $\ln$ or $\exp$. Find all applications $f$ from $\mathbb{R}\rightarrow\mathbb{R}$ such that (1) is satisfied, posing all the necessary restrictions.
Thank you very much for any help, or pointing me out if the question is a duplicate.