A real number raised to a real exponent is defined by
Prove the following properties:
- .
- .
- .
- .
- For , if and only if .
- For all of these we use the definition and then use the corresponding properties of the exponential function . (These are proved for the function in Theorem 6.6, and then we define in Section 6.14.)
- Proof.
- Proof.
- Proof.
- Proof.
- Proof. Assume . For the forward direction, assume . We have
Therefore, since we have , and so
Conversely, if then we have