Let me state how I define exponentials.

Define

It is not hard to show that,

1)

is an increasing function.

2)

is differenciable and therefore continous for

.

3)

.

4)There exists a number, called

, so that

.

We see that

is a one-to-one function. Define

to be its inverse function on its range.

It is not hard to show that

is increasing, differenciable and so continous, and furthermore,

.

We have the following supprising property that if

is a

*rational* number then

. So we

*define* , to be

. With that we generalize exponents as follows

.

The difficutly besides for using the definition is to have a formal definition of what an exponent means.