Properties of Exponents

Product Rule for Exponents

For integer exponents, we defined to mean “a used as a factor x times.” Since this definition extends naturally to all real exponents, we see that .

Quotient Rule for Exponents

Similarly, .

Power Rule for Exponents

Using the same definition, we have .

Some Useful Properties

It follows from these that , , and .

Recall from the definitions that .

A final useful property of exponents is .

To derive this identity, let . If we take the base of a logarithm of both sides, we have . Using the exponent rule for logarithms, we have. Since , we know . Since the base a logarithm is a one-to-one function, we may deduce that y = x; that is,.