Exercises 7880 develop an elegant approach to the exponential and logarithm functions
Chapter 5, Problem 79(choose chapter or problem)
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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