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**ThePerfectHacker** Let me state how I define exponentials.

Define $\displaystyle \ln x = \int_1^x \frac{1}{t} dt \mbox{ for }x>0$

It is not hard to show that,

1)$\displaystyle \ln x$ is an increasing function.

2)$\displaystyle \ln x$ is differenciable and therefore continous for $\displaystyle (0,\infty)$.

3)$\displaystyle (\ln x)' = \frac{1}{x} \mbox{ for }x>0$.

4)There exists a number, called $\displaystyle e$, so that $\displaystyle \ln e = 1$.

We see that $\displaystyle \ln x$ is a one-to-one function. Define $\displaystyle \exp x$ to be its inverse function on its range.

It is not hard to show that $\displaystyle \exp x$ is increasing, differenciable and so continous, and furthermore, $\displaystyle (\exp x)' = \exp x$.

We have the following supprising property that if $\displaystyle q$ is a *rational* number then $\displaystyle \exp q = e^q$. So we *define* $\displaystyle e^r$, to be $\displaystyle \exp r$. With that we generalize exponents as follows $\displaystyle a^x = e^{\ln a x} = \exp (\ln a x) \mbox{ for }a>0$.

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