Let me state how I define exponentials.
It is not hard to show that,
1) is an increasing function.
2) is differenciable and therefore continous for .
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.