## Quick Reference

Let *a* be a positive number not equal to 1. The exponential function to base *a* is the function *f* such that *f*(*x*)=*a*^{x} for all *x* in **R**. This must be clearly distinguished from what is commonly called ‘the’ exponential function. The graphs *y*=2^{x} 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 *a*^{x} can be done in two ways:

1. The familiar rules for indices (see index) give a meaning to *a*^{x} 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 *a*^{1.4}, *a*^{1.41}, *a*^{1.414},…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: *a*^{x}=exp(*x* ln *a*). This approach is less elementary, but really more satisfactory than **1**. It follows that ln(*a*^{x})=*x* ln *a*, as would be expected, and the following can be proved: *a*^{x+y}=*a*^{x}, *a*^{y}, *a*^{−x}=1/*a*^{x} and (*a*^{x})^{y}=*a*^{xy}.*n* is a positive integer, *a*^{n}, defined in this way, is indeed equal to the product *a*×*a* ×⋯× *a* with *n* occurrences of *a*, and *a*^{1/n} is equal to^{n}√a.

*a*^{x+y}=*a*^{x}, *a*^{y}, *a*^{−x}=1/*a*^{x} and (*a*^{x})^{y}=*a*^{xy}.

*n* is a positive integer, *a*^{n}, defined in this way, is indeed equal to the product *a*×*a* ×⋯× *a* with *n* occurrences of *a*, and *a*^{1/n} is equal to^{n}√a.

*Subjects:*
Mathematics.