Let b > 0 and let (x) = F(xG(b)) with F as in Exercise 117. Use Exercise

Let b > 0 and let ƒ(x) = F(xG(b)) with F as in Exercise 117. Use Exercise 116 (f) to prove that ƒ(r) = b^r for every rational number r. This gives us a way of defining b^x for irrational x, namely b^x = ƒ(x). With this definition, ƒ(x) = b^x is a differentiable function of x (because F is differentiable).

Data From Exercise 117

Use Exercise 116 to prove the following statements.

(a) G has an inverse with domain R and range {x : x > 0}. Denote the inverse by F.
(b) F(x + y) = F(x)F(y) for all x, y. It suffices to show that G(F(x)F(y)) = G(F(x + y)).
(c) F(r) = Er for all numbers. In particular, F(0) = 1.
(d) F(x) = F(x). Use the formula for the derivative of an inverse function.
This shows that E = e and that F(x) is the function ex as defined in the text.

Data From Exercise 116

G(a^r) = rG(a) for all a > 0 and rational numbers r.