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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.

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    $\begingroup$ Is there a function $\mathbb R \to \mathbb R$ satisfying $(1)$? [ I note that logarithms are only $(0,\infty) \to \mathbb R$.] Let's see ... $f(0)=f(0\cdot y) = f(0)+f(y)$ so $f(y) = 0$ for all $y$. Easy. $\endgroup$ Dec 25, 2023 at 15:55
  • $\begingroup$ @GeraldEdgar thanks but I thought I was clear frm this fractional relation the ln imposing all the necessary restrictions. $\endgroup$
    – Dimitris
    Dec 25, 2023 at 16:33
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    $\begingroup$ Essentially all you need to do is assume continuity, and you will have a logarithm math.stackexchange.com/questions/2710603/… Also see en.wikipedia.org/wiki/Cauchy%27s_functional_equation $\endgroup$
    – user52817
    Dec 25, 2023 at 21:26
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    $\begingroup$ This question should be migrated to math.se $\endgroup$ Dec 30, 2023 at 17:36
  • $\begingroup$ Re "thanks but I thought I was clear frm this fractional relation the ln imposing all the necessary restrictions" It's not clear at all what "all the necessary restrictions" mean, and the domain of the function is extremely relevant to determine the set of solutions. $\endgroup$
    – Stef
    Jan 3 at 13:21

1 Answer 1

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Find all applications $f$ from $\mathbb{R} \to \mathbb{R}$ such that (1) is satisfied, posing all the necessary restrictions.

From the textbook Calculus 9th edition by Salas, Hille, Etgen (section 7.2):

From our point of view the fundamental property of logarithms is that they transform multiplication into addition: the log of a product = the sum of the logs. Taking this as the central idea, we are led to a general notion of logarithm [...]

DEFINITION 7.2.1 A logarithm function is a nonconstant differentiable function $f$ defined on a set of positive numbers such that for all $a > 0$ and $b > 0$ , $f(ab) = f(a) + f(b)$.

Let's assume for the time being that such logarithm functions exist, and let's see what we can find out about them.

After some work they get $f'(x) = \dfrac{1}{x} f'(1)$.

The most natural choice, the one that will keep calculations as simple as possible, is to set $f'(1)=1$.

more discussion

This function, which takes on the value 0 at 1 and has derivative $1/x$ for $x>0$, must, by Theorem 5.3.5, take the form $\int_1^x \dfrac{dt}{t}$.

DEFINITION 7.2.3 The function $L(x) = \int_1^x \dfrac{dt}{t}, x > 0$, is called the (natural) logarithm function.

Then pages of developing the properties of $L(x)$ before defining the exponential function in section 7.4.


Edit (addendum):

THEOREM 5.3.5 Let $f$ be continuous on $[a,b]$. The function $F$ defined on $[a,b]$ by $F(x) = \int_a^x f(t) dt$ is continuous on $[a,b]$, differentiable on $(a,b)$ and has derivative $F'(x) = f(x)$ for all $x$ in $(a,b)$.

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    $\begingroup$ Do you want to see how they got from $f(ab)=f(a)+f(b)$ to $f'(x)=\dfrac{1}{x} f'(1)$? $\endgroup$
    – Marlene
    Dec 26, 2023 at 22:56

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