equals a constant. The constant can only be 0 because f

(1)

= 0. This proves the
last formula of 8.13 and completes the proof of the theorem. ■

The last formula tells how to define x^{α} for any x > 0 and α ∈ ℝ. I want to stress this
is something new. Students are often deceived into thinking they know what x^{α}
means for α a real number. There is no place for such deception in mathematics,
however.

Definition 8.3.13Define x^{α}for x > 0 and α ∈ ℝ by thefollowing formula.

ln(xα) = αln(x).

In other words,

xα ≡ exp (αln(x)).

From Theorem 8.3.12 this new definition does not contradict the usual definition in the
case where α is an integer.

From this definition, the following properties are obtained.

Proposition 8.3.14For x > 0 let f

(x)

= x^{α}where α ∈ ℝ. Then f^{′}

(x)

= αx^{α−1}.Also x^{α+β} = x^{α}x^{β}and

(xα)

^{β} = x^{αβ}.

Proof: First consider the claim about the sum of the exponents.