The “natural base,” e
In nature, many populations grow exponentially, in a manner that is best modeled by a real number which we call e. This number is defined as 
.
Example
The population of bacteria in a certain culture at time t is given by 
, where t is measured in hours. Thus 100 bacteria are initially present, 
bacteria are present after 10 hours, and roughly 
bacteria are in the culture after 100 hours.
Note that on some calculators and computer algebra systems, e is denoted exp. In these systems, the function 
will usually be input as exp(x), but check your user manual for correct syntax.
Students of calculus will recall that the function is remarkable because, for all real values of x, the derivative of ex is ex. This means that the rate of change of the function at time x=t is the same as f(t).