Why natural log has base e?

You can have logaritms(logs) to any 'base'. However, modern calculators are programmed to base '10' (log button) or base '2.718281828....' (an irrational number) (ln button).

The number '2.71828....' is known as the Exponential or Natural number.

Hence on a calculator the 'ln' button is known as the exponential.

If you draw a graph using '2.71828...' as the 'power/index/exponential, the graph curve would very quickly go 'off the top of the graph' paper.

e.g.

0^(2.718...) = 1

1^(2.718..._ = 2.71828...

2^(2.71828...) =7.389....

3^(2.71828....) = 20.085536....

As you can see all the results are increasing rapidly, and exponentially, and the answers are 'irrational.