Exponential functions are closely related to geometric sequences. A geometric sequence of numbers is one in which each successive number of the sequence is obtained by multiplying the previous number by a fixed factor m. An example is the sequence {1, 3, 9, 27, 81, …}. If we label the numbers in the sequence as {y0, y1, y2, …} then their values are given by the formula:yn = y0 · m n.The geometric sequence is completely described by giving its starting value y0 and the multiplication factor m. For the above example y0 = 1 and m = 3. Another example is the geometric sequence {40, 20, 10, 5, 2.5, …} for which y0 = 40 and m = 0.5.
The exponential function is simply the generalization of the geometric sequence in which the counting integer n is replaced by the real variable x. We define an exponential function to be any function of the form:y = y0 · m x.It gets its name from the fact that the variable x is in the exponent. The “starting value” y0 may be any real constant but the base m must be a positive real constant to avoid taking roots of negative numbers.
The exponential function y = y0 · m x has these two properties:When x = 0 then y = y0.
When x is increased by 1 then y is multiplied by a factor of m. This is true for any real value of x, not just integer values of x. To prove this suppose that y has some value ya when x has some value xa . That is:Now increase x from xa to xa+1. We get:
We see that y is now m times its previous value of ya. If the multiplication factor m > 1 then we say that y grows exponentially, and if m < 1 then we say that y decays exponentially.
