The Number e

e is a number that can be defined in many ways. First,

e = 1+    

Also, e is the number such that

= 1    

The numerical value of e is approximately 2.71828.... The function f (x) = ex is an exponential function, which is a function of the form f (x) = ax, where a is a positive constant. The graph of f (x) = ex is shown below.

Figure %:Graph of the function f (x) = ex

The Natural Log

The inverse of the function ex is the natural log function ln(x). These two graphs are pictured below:

Figure %:ex and ln(x) are inverses

Recall that if logab = x, then x is the power that a must be raised to in order to equal b.

With the natural log, the base is e. So, for example, ln(e) = 1. Because ex and ln(x) are inverses, the following relations hold: