Exponential Functions

Given a positive real number b and a rational number m/n, the rational power b^m/n is defined as

the positive nth root of b^m. The negative power b^-m/n is

As an example consider b = 10. Several values of 10^m/n are shown in Table 2.5.2.

Table 2.5.2.

If we plot all the rational powers 10^m/n, we get a dotted line, with one value for each rational number m/n, as in Figure 2.5.6.

By connecting the dots with a smooth curve, we obtain a function y = 10^x, where x varies over all real numbers instead of just the rationals. 10^x is called the exponential function with base 10. It is positive for all x and follows the rules

10^a+b = 10^a · 10^b, 10^a·b = (10^a)^b.