Do u know why we call 'e' as natural no??

e was invented by Leonard Euler, and is sometimes called Euler's constant, but it's proper name is still 'e'. Euler is pronounce "Oiler".

Take the formula a^x, where a is some number. Say 2^x.

2^0 is 1

2^1 is 2

2^2 is 4

2^3 is 8

and so on.

You can draw this on a graph. It starts fairly flat but rises very quickly and quickly goes off the top of the paper.

You can draw a tangent line to this curve (a line which touches the curve). The steepness or slope of this line measures the rate at which the formula is increasing at the place where the line touches the curve.

A remarkable thing happens. For the 2^x graph, at every point on the graph, the rate of increase of 2^x is 0.693 2^x. The rate is proportional to the height of the graph. Positive feedback!

Try the same thing with 10^x. The rate of increase is 2.3 10^x

Same idea.

So if 2^x rises at 0.693 2^x which is less than 2^x

and 10^x rises at 2.3 10^x which is more than 10^x

then is there a number a between 2 and 10 for which

a^x rises at exactly a^x?

Yes there is. That number is e.

You can take logs to any base, but no base seems better than any other. Is there a base which is somehow more natural than 2 or 10? Yes: base e.

The natural log of x is written ln(x)

In my examples above, the mysterious 0.693 is ln(2)

the mysterious 2.3 is ln(10)

Because e^x is that magic function whose rate is equal to itself, it pops up all over the place where there are problems to do with rates.

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