# e    Natural logarithms    e

Interesting numbers --- zero --- one --- complex --- root 2 --- golden ratio --- e --- pi --- googol --- infinity

### What is e?

The constant e is an irrational number. It has this property:

 d(ex)dx = ex

ex means this number, e, raised to the power of the variable x.
dy/dx means the differential of y with respect to x. If you draw a graph of y against x, then the differential of y with respect to x means the gradient of the graph at point (x,y). For example, if y=x2, then dy/dx=2x. But here y=ex, and dy/dx is the same as y, which is a bit weird.

The exact reasons for the use of the letter e are unknown, but it may be because it is the first letter of the word exponential.

### Value of e

e = 2.71828 18284 59045 23536 (truncated to 20 decimal places)

The formula for working out the value of e looks fun.

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + ....

The ! symbol means factorial. It's sometimes known as 'shriek'! 5! means 1x2x3x4x5. 1! is just 1. 0! is defined as 1 as well. Click on Calculate e to use this formula to calculate e to more and more accuracy. All computer calculations are done to a particular precision, so there is a limit to how accurate a value for you can get for e with this webpage. But see how accurate you can get.

Number of terms of formula:

e (approx) =

### Logarithms

Logarithms were invented by John Napier,a Scottish Mathematician, in 1614. Click here to see how log tables (base ten) work. (John Napier also invented Napier's bones but those have nothing to do with logarithms or e.)

You can have logarithms to other bases than ten. In fact, John Napier's original logarithms were to base e. It may seem odd to have logarithms to a base which is not a whole number (and in fact is irrational), but it is easier to calculate a natural logarithm. Someone has to calculate logarithms or we wouldn't have log tables!

natural log of x or ln(x) = (x - 1) - (x - 1)2 / 2 + (x - 1)3 / 3 - (x - 1)4 / 4 + (x - 1)5 / 5 - (x - 1)6 / 6 + ....

This formula goes on for ever, so what good is it? Luckily, we don't need all of it. We can work out the log of x between zero and two, so x2 is smaller than x, and higher powers of x get smaller and smaller. This means that we can get as many significant figures of the logarithm that we want, then stop and ignore the rest of the formula. You can calculate a natural log below. Choose a number between zero and two, or use the value given (0.5). Carry on clicking on Calculate natural log to get more accurate values. You'll see that this formula is a lot slower than the formula for e.

Enter a number between zero and two:

Number of terms of formula:

Log tables sometimes use base 10 rather base e. So how do we get logarithms to base 10.? There's another formula.

Log base 10 of x = Natural log of x / Natural log of 10

where Natural log of ten = 2.30258 50929 94046 to 15 decimal places

### Formula using e

Euler's formula states that      eix; = cos x + i sin x

The special case where x = π(180°) is      e = -1

This formula links e, π and i !