Exercises 7880 develop an elegant approach to the exponential and logarithm functions

Chapter 5, Problem 79

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Exercises 7880 develop an elegant approach to the exponential and logarithm functions. Define a function G(x) for x > 0: G(x) = x 1 1 t dt Defining ex Use Exercise 78 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. Hint: 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). Hint: Use the formula for the derivative of an inverse function that appears in Exercise 28 of the Chapter 3 Review Exercises. This shows that E = e and F (x) = ex as defined in the text.

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