Why use natural logarithm $e$?

Question

I have a question regarding the natural logarithm $e$.

Simply, why is $e$ special enough to:

1. Have its own special notation $\ln$?
2. Be used in derivatives?
3. Have Wikipedia pages dedicated to it?

From my current understanding, it is simply a non-reoccuring irrational number. So what makes it apart from other irrational numbers such as pi?

Answer 1

Suppose we have a function like below. Just call it $F(x)$ for now rather than any fancy name.

$$ y = F(x) = \int_1^x\frac{1}{t}dt $$

From the fundamental laws of calculus, we can easily conclude that:

$$ \frac{dy}{dx} = F'(x) = \frac{1}{x} --- (1) $$

And with a bit effort we can conclude:

$$ F(x^r) = rF(x) --- (2) $$

Now let's say the reverse function of $F(x)$ is $G(x)$:

$$ x = G(y) = F^{-1}(y) $$

We can easily conclude that the derivative of $G(y)$ is:

$$ G{'}(y) = \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{\frac{1}{x}} = x = G(y) --- (3) $$

We can write $G(y)$ to $G(x)$ and it has a unique behaviour that:

$$ G'(x) = G(x) $$

Now let's do some calculation about $a^x$. Since $G$ and $F$ are reverse function to each other. We have:

$$ a^x = G(F(a^x)) $$

Then let's continue the calculation with the help of (2):

$$ a^x = G(F(a^x)) = G(xF(a)) $$

Only when $F(a) = 1$ can we have:

$$ a^x = G(x) --- (4) $$

To make $F(a) = 1$, $a$ must be $2.71828...$. Euler called this constant $e$.

And if we call $F(x)$ as $ln(x)$. And $G(x)$ as $exp(x)$, we have:

(1) => $ln'(x) = \frac{1}{x}$

(2) => $ln(x^r) = rln(x)$

(3) => $exp'(x) = exp (x)$

(4) => $e^x = exp(x)$

and (3), (4) =>

(5) $(e^x)' = e^x$

So from the above construction and deduction, we can see that $e$ is a special value that is determined by $F(x) = 1$.

And with such a value we can define some nice functions that have nice behaviors. i.e. $ln(x)$ and $exp(x)$ or $e^x$.

We construct these functions through the calculus operations. Besides the nice differential behaviors, these functions also have some similar behaviors to the conventional logarithm/exponent functions, which I didn't list here. I think that's why they are given the fancy names $ln$ and $exp$.