Originally Posted by

**HallsofIvy** Quite frankly, before I saw "using the intermediate value theorem" I was going to ask "what are your definitions for e and ln?"

One way of defining e is to define "$\displaystyle a^x$" for all real numbers x and all positive numbers a, define "e" as the number such that $\displaystyle \displaytype\lim_{x\to 0}\frac{e^x- 1}{x}= 1$, then define ln(x) as the inverse function to $\displaystyle e^x$.

Another way is to define $\displaystyle ln(x)= \int_1^x \frac{dt}{t}$ and then define $\displaystyle e^x= exp(x)$ as **its** inverse function.

In both of those cases, ln(e)= 1 follows immediately from the definition of "inverse function".