What is natural about the natural logarithms ?

Question

Why are logarithms to base e natural?

Answer 1

e has the property that d/dx e^x = e^x. In other words, it's its own derivative.

Answer 2

Abe and Jac are correct, but they don't really give you a good understanding about this number e and what makes it so special that it just seems "natural" to use it as the basis for logarithms.
So I'll attempt to say it clearly.

Suppose you had a function f(x) that had this characteristic:
If you make a graph of f(x) and then check the slope of the graph at various points, you discover that the slope is always equal to the value of f(x)!
If f(x) = 1, then the slope at that point is 1. If you try a different value of x, and find that f(x) = 2, then the slope at that point is 2.
If there is a point where f(x) = 0.1, then the slope at that point is 0.1. Or if f(x) = 0, the slope is 0.
And if at any point f(x) = infinity, then the slope is infinite.

It turns out that there is a function with those characteristics, and it is an exponential function (like 10^x or 2^x). The particular function that has this characteristic is f(x) = e^x, where e has been found to be a value of approximately 2.718281828....
(It's easy to remember this approximation to 9 decimal places, because those 4 digits 1828 repeat themselves (not over and over, just one repeat).

Because the rate of change (slope) of this function is the same as the value of the function, we have f'(x) = f(x) = e^x. And it follows that the second derivative is ALSO equal to the first derivative and the function: f''(x) = f'(x) = f(x) = e^x. In fact, derivatives of ALL orders are equal to e^x.

Similarly, if we use e as the base for logarithms, then the slope of the log function has a very simple form: 1/x.
That is, d/dx (ln x) = 1/x.

(Actually, that should be ln (absolute value of x), which is important if you want to use negative x values. And it's even more important if you want to determine the integral of 1/x, which is ln (ABS(x)), not simply ln x.)

Hope that makes the "naturalness" of e somewhat clearer to you.

Using logs base 10 used to have the important characteristic that if you knew the log of a number you could find the log of a number 10 times as large by adding 1 to the log of the first number. Base 10 logs still have that characteristic, but it is no longer important, since we have calculators and computers to calculate logs. It's hard to think of a reason to use base 10 logs today. But natural logarithms tend to occur naturally, and are easy to work with in formulas.

Answer 3

Initially, it seems that in a world using base 10 for nearly all calculations, this base would be more "natural" than base e. The reason we call the ln(x) "natural" is twofold: first, expressions in which the unknown variable appears as the exponent of e occur much more often than exponents of 10 (because of the "natural" properties of the exponential function which allow it to describe growth and decay behaviors), and second, because the natural logarithm can be defined quite easily using a simple integral or Taylor series--which is not true of other logarithms. Thus, the natural logarithm is more useful in practice.

Answer 4

I can't really give a better answer than Actuator's which is a really elegant description of why base e is so special.

One other link between 'natural' and 'e' is the following. (This not why natural logs are so called, but it is just an interesting link).

Measures taken from many processes in nature are distributed according to a Normal Distribution (alias Gaussian Distribution, or Bell Curve). For instance, if you were to plant a set of bean seeds and then measure the height of the plants after a certain period of time the heights would form an approximately normal distribution. And, surprise, surprise, the Normal Distribution is described by a very simple function of e.

It turns out that the Normal Distribution crops up so often relates to mathematical properties of the distribution itself - in fact to 'e' itself.

So compelling is 'e' that you might say it is a discovery rather than invention - and that too suggests using the word 'natural'.

Answer 5

Natural processes like growth or decay (number of atoms, size of population) are functions of e to a given power.

natural logarithms are logs to the base of e.

Answer 6

base e itself represents natural numbers, so any logarithm to base of a natural number will be a natural number.

Answer 7

Just a quick bit of trivia:
Natural logs are also known as Napierian logs after the Scottish mathematician, John Napier who discovered them!

Answer 8

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