What's so natural about the natural logarithm?

Question

I've always heard that this magic number e is ever-present in nature, that there's something fundamental and primal about it, but I have never understood why. I'd like to understand e, understand y = e^x, and understand y = ln x in the same way I understand pi (which, by the way, I conceptualize as "the ratio of circumference to diameter"). The best answer will give me some simple concept like that (if such a thing is possible) and also explanations and examples. Thank you in advance for your efforts!

Zanti-- it might be said that pi demonstrates the same thing (inasmuch as we accept that circles are "natural").

Keith -- I scrupulously ignored calculus during my senior year of high school, so thank you for any detail you add.

Answer 1

Any number raised to the power of zero equals 1:

1^0=1
2^0=1
3^0=1

Now consider a series of functions:
f(x)=1^x
f(x)=2^x
f(x)=3^x

It is clear that when x is zero, all of these functions =1; that means that all of these functions pass through the point (0,1). But what about the *slope* of these functions? They're all different. All the graphed lines of these functions pass through the point (0,1) but at different slope angles.

The "natural" point of the number e is this:

when the function e^x passes through the point (0,1), the slope of the function is exactly 1.

And, even more generally, when the function f(x)=e^x reaches any value, its slope is the same as that value. In calculus terms, the function f(x)=e^x is its own derivative.

Answer 2

Maybe natural is not a very accurate adjective, but one of the reasons is that the definition of the exponencial function a^x, a>0, x in R, is based on the number e. First, you define e and the function e^x, and then you define a^x = e^(x ln(a)), where ln is the natural logarithm. So, whenever you see, no matter where, an exponencial function, you implicitly see the number e.

And another reason e is called natural is that the derivative of f(x) = e^x is f itself.

The number e is given by the limit of the series e = 1 + 1 + 1/2! +....1/n!. We know e is not only irrational but transcendent, that is, no polynomial with rational roots has e as one of its roots. The number pi is transcendent, too.

Answer 3

e is a constant with the value of 2.71828... and can be approximated by:
summation notation from term 0 to infinity of the rule (1/n!)
e is also used for calculating money or some type of population over time.

Answer 4

Yes, well, since e is an irrational number, I think this proves that irrationality is natural. :)