Ii the range of exp a is the set of positive real

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Chapter 0 / Exercise 3
Mathematical Applications for the Management, Life, and Social Sciences
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(ii) The range of exp a is the set of positive real numbers. (iii) The function exp a is continuous. (iv) The function exp a converts addition into multiplication, i.e. exp a ( x + y ) = exp a ( x ) · exp a ( y ) . (v) The value of exp a ( x ) when x = 1 is exp a (1) = a. 7 My favorite is Stewart: Calculus: Early Transcendentals . 53
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Chapter 0 / Exercise 3
Mathematical Applications for the Management, Life, and Social Sciences
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§ 16.2. The more familiar notation for the exponential function is exp a ( x ) = a x so parts (iv) and (v) of Theorem 16.1 take the more familiar form a x + y = a x · a y , a 1 = a. ( ) This implies that, for integers, exponentiation is repeated multiplication, 8 e.g. a 3 = a 1+1+1 = a 1 · a 1 · a 1 = a · a · a . Using ( ) repeatedly gives a nx = a x + x + · · · + x n = a x · a x · · · · a x n = ( a x ) n so taking x = 1 /n and using ( ) again proves that for any positive integer n a 1 n = n a the n th root of a (i.e. the functions b = a n and a = b 1 n are inverse functions). It follows easily that parts (iv) and (v) of Theorem 16.1 determines the value a x uniquely when x is a rational number 9 and, as every real number is a limit of rational numbers, this shows that the conditions of Theorem 16.1 uniquely determine the exponential function. In Section 38 we give a formula for exp a and show that it satisfies the conditions of Theorem 16.1. § 16.3. The following familiar laws of algebra all follow easily from equation ( ). a 0 = 1 , a x + y = a x · a y , a - x = 1 a x , a 1 = a, ( ab ) x = a x · b x , ( a p ) x = a p · x . For example, the reason why ( ab ) x = a x · b x is that both sides satisfy the conditions of Theorem 16.1 (reading ab for a ). Similarly, to prove ( a p ) x = a p · x note that both sides (as functions of x and reading a p for a ) satisfy the conditions of Theorem 16.1 (at least if p = 0; if p = 0 both sides equal 1). 8 Just as multiplication is repeated addition. 9 A rational number is a ratio of integers, i.e. a fraction. 54
Theorem 16.4. For a > 1 , the exponential function is differentiable, strictly increasing, and satisfies so lim x →∞ a x = , and lim x →-∞ a x = 0 . These will be proved in section 38. That the exponential function y = a x is strictly increasing means that a x 1 < a x 2 for x 1 < x 2 . (For example, 2 3 < 2 4 .) This, combined with the Intermediate Value Theorem 9.7 means that the range of the exponential function is the set of all positive numbers and the graph of the exponential function passes the vertical line test so the exponential function has an inverse function. Definition 16.5. The inverse function to the exponential function exp a ( x ) = a x is called the logarithm function base a : y = a x ⇐⇒ x = log a ( y ) . The range of log a is all real numbers, the domain is all positive rea numbers, and log a ( a x ) = x, a log a ( y ) = y by the Cancellation Law for inverse functions. Remark 16.6. Since (1 /a ) x = 1 /a x = a - x , similar statements hold if a < 1. Thus for b < 1 lim x →∞ b x = 0 and lim x →∞ b - x = and the function b x is strictly decreasing.
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