We have cited polynomial functions, rational functions, and trigonometric functions as belonging to the class of elementary functions. In this section we add the exponential functions and logarithmic functions to the list. And you will be happy to know that that is close to the end of the story. We will see that much of calculus is involved in studying the properties of the basic functions and in learning to use them in applications that rely upon their unique properties. The full class of elementary functions provides a rich source from which we can draw functions to represent a variety of real world objects and model their behavior.
Exponential functions make their most dramatic debut in population modeling.
Beginning with a population of 67.38 million people in 1980, the population increased each year by a factor of 1.026. Thus in 1981 the population was million, in 1982 it was million, and in general it is where is the number of years that have elapsed since 1980. This is an exponential function. It is called an exponential function because the base is constant , in this case the constant 1.026, and the independent variable is in the exponent. The base represents the growth factor by which the population increases each year.