e (idea)

The definition

By definition, for all real numbers x, f(x)=e^x is the function whose rate of change (derivative) is equal to f(x), i.e., f'(x)=f(x) for each x in the real numbers. Thus a Taylor polynomial would be easy to construct (and use to approximate f(x)), if we were given any specific information about the value of f(x) for some real number x. So let f(0)=a, an arbitrary (non-zero) real number. Then the nth Taylor polynomial is:

                                                              (n)
 (n)                f'(0)       1    f''(0)       2          f   (0)       n
T   (x,0) = f(0) + -------·(x-0)  + --------·(x-0)  + ... + ---------·(x-0)
                      1!               2!                       n!

               a       1    a       2          a       n
        = a + ---·(x-0)  + ---·(x-0)  + ... + ---·(x-0)
               1!           2!                 n!

               n
              --- /  i \
              \   | x  |
        = a · /   | -- |
              --- \ i! /
              i=0

Now, it would be nice if f(0)=1, because then a=1 and there are no extra constants floating around. Notice that a=0 is possible, but highly uninteresting, since then the function and all its derivatives are zero. It is also worth noting that the above formula is only an approximation; the real value of f(x) can only be determined by adding up the countable number of terms in the sequence.

An alternate definition

e^x can also be defined as the inverse of ln(x), the natural logarithm of x. All the properties of powers of numbers apply to e^x, which can be easily shown using the definition of ln(x).