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

Chapter 5, Problem 78

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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 ln x as an Integral This exercise proceeds as if we didnt know that G(x) = ln x and shows directly that G has all the basic properties of the logarithm. Prove the following statements: (a) ab a 1 t dt = b 1 1 t dt for all a, b > 0. Hint: Use the substitution u = t/a. (b) G(ab) = G(a) + G(b). Hint: Break up the integral from 1 to ab into two integrals and use (a). (c) G(1) = 0 and G(a1) = G(a) for a > 0. (d) G(an) = nG(a) for all a > 0 and integers n. (e) G(a1/n) = 1 n G(a) for all a > 0 and integers n = 0. (f) G(ar) = rG(a) for all a > 0 and rational numbers r. (g) G is increasing. Hint: Use FTC II. (h) There exists a number a such that G(a) > 1. Hint: Show that G(2) > 0 and take a = 2m for m > 1/G(2). (i) lim x G(x) = and lim x0+ G(x) = . (j) There exists a unique number E such that G(E) = 1. (k) G(Er) = r for every rational number r.