The definition of $\ln(x)$

Question

When I taught my student the logarithm, he asked me about the historical definition of $\ln(x)$.

1. The first definition I found is that $$\ln(x)=\int_{1}^{x}{ \frac{dt}{t} } $$
2. Defined as the logarithm to base $e$ or the inverse function of the exponentiation to base $e$: $$\ln(x)=y \Longleftrightarrow e^y=x$$ where $e$ defined as $$e=\lim_{n\to\infty}\left( 1+\frac{1}{n} \right)^n$$

Which is the real definition of the logarithm?

Answer 1

I'm no expert on maths history, but logarithms are old enough not to have a "historical definition" that meets our standards of what a definition should be. I think the integral definition of the logarithm is the better one to teach, for a few reasons:

1. What is exponentiation? Even if you define $e$, that may instantly tell you $e^2$ or $e^{-\frac{1}{3}}$, but how do you tell what $e^{\pi}$ is? No combination of repeated multiplication, inversion, or taking roots of $e$ will produce this number. (Note, this can be mediated by defining $\exp(x) = \lim_{n\to\infty} \left(1 + \frac{x}{n}\right)^n$).

2. The indefinite integral of $\frac{1}{x}$ is such a natural question that it warrants the invention of a function to fill the gap.

3. The log laws (and hence exponential laws) turn into lovely applications of various integral rules.

4. The calculus properties of $\ln$ and $\exp$ follow immediately from this definition too.

That's why I would teach the integral definition.

Answer 2

The fact that they call it a "logarithm" implies the must have had a concept that it is the logarithm of some base. So when the defined they must have been using the concept $\ln x = y \iff e^y = x$. And I even imagine they would be aware that $\frac {db^x}{dx} = C_b*b^x$ (for rational values of $x$; irrational values would have been poorly understood) so that would figure there must be a base so that $C_b = 1$ and $\frac {de^x}{dx} = e^x$.

But although that can be the concept and germination of a definition, it can't actually be a practical definition until after they had some way of finding what $e$ would be. And I imagine to do that they had to recognize that $\int \frac 1x dx$ is a logrithmic function and the value of it's base would be $\lim (1 +\frac 1n)^n$.

So I would guess, it went in this order 1) $\frac {db^x}{dx} C(b)*b^x$ for some function $C(b)=\lim\frac {b^h - 1}h$. 2) That therefore $C(b) = \int_1^b \frac 1t dt$ and that $C(b)$ 3) the $e$ so that $C(b) = 1$ is $\lim(1 + \frac 1n)^n$ then 4) noting $\log_e (x) = C(x)$ is an immediate consequence and then the final definition 4) $\ln x := \log_e x = \int_1^x\frac 1t dt = C(x)$.

But I'm just guessing.