Let a be a positive number not equal to 1. The exponential function to base a is the function f such that f(x)=ax for all x in R. This must be clearly distinguished from what is commonly called ‘the’ exponential function. The graphs y=2x and y = (½)x illustrate the essential difference between the cases when a>1 and a < 1. See also exponential growth and exponential decay. Clarifying just what is meant by ax can be done in two ways:
1. The familiar rules for indices (see index) give a meaning to ax for rational values of x. For x not rational, take a sequence of rationals that approximate more and more closely to x. For example, when x=√2, such a sequence could be 1.4, 1.41, 1.414,…. Now each of the values a1.4, a1.41, a1.414,…has a meaning, since in each case the index is rational. It can be proved that this sequence of values has a limit, and this limit is then taken as the definition of a√2. The method is applicable for any real value of x.
2. Alternatively, suppose that exp has been defined (say by approach 2 to the exponential function) and that ln is its inverse function. Then the following can be taken as a definition: ax=exp(x ln a). This approach is less elementary, but really more satisfactory than 1. It follows that ln(ax)=x ln a, as would be expected, and the following can be proved: