The number e
There exists an irrational number that is not represented with a number or a symbol (like ), but rather is represented by the letter e.
If you use the e key on your calculator it will give you a decimal approximation of 2.718281828. However, this is only an approximation. Because e is an irrational number, it cannot be completely and accurately represented with a decimal.
Other than using the calculator, the number e can also be approximated with the exponential expression . As you use larger and larger values of x, the exponential expression gets closer and closer to e. x   10  2.59374  100  2.70481  1000  2.71692  10000  2.71815  100000  2.71827  1000000  2.71828 
Many applications involve using an exponential expression with a base of e. Applications of exponential growth and decay as well as interest that is compounded continuously are just a few of the many ways e is used in solving real world problems. Because it is treated as a number (and not as a variable), all the rules of exponents apply to e as it does any other exponential expression. It can also be graphed as other exponential functions.
The graph of is shown below.   You can see that it has the same characteristics of other exponential functions. 
The natural logarithm (ln)
Another important use of e is as the base of a logarithm. When used as the base for a logarithm, we use a different notation. Rather than writing we use the notation ln(x). This is called the natural logarithm and is read phonetically as “el in of x”.
Just because it is written differently does not mean we treat it differently than other logarithms. We can graph it the same as any other logarithmic function (link to logsgraphing.doc not yet written)
The graph of is below.   Notice that it has the same basic properties as other logarithmic graphs. 
And as with other logarithmic graphs, the graph of is the inverse of the exponential function . In fact, if you look at the characteristics listed for the two graphs, you’ll see the x and y have been interchanged.
The natural logarithm follows the same properties as other logarithms. However, since the natural log is written slightly different, it may be helpful to see the properties written in ln form.

Examples 

You may wish to do the practice problems below to make sure you are able to evaluate e on your calculator and use the properties of the natural logarithm. Each of these solutions is found by typing the problem directly into the calculator. Answers have been rounded to three decimal places. Note: to obtain a display of the numerical value of e on your calculator use the keystrokes 




