517 CHAPTER 4 Test Prep Concepts Examples (a) 2x is a unique real number for all real numbers x. (b) 2x = 23 if and only if x = 3. (c) 25 6210, because 2 71 and 5 610. (d) A1 2B 5 7 A1 2B 10 , because 0 61 2 61 and 5 610. Graph ƒ1x2 = 3x. Give the domain and range. x y -1 1 3 0 1 1 3 The domain is 1-∞, ∞2, and the range is 10, ∞2. 0 1 y 2 4 f(x) = 3x (1, 3) (0, 1) x Q–1, R 1 3 4.2 Exponential Functions Additional Properties of Exponents For any real number a 70, a≠1, the following hold true. (a) ax is a unique real number for all real numbers x. (b) ab = ac if and only if b = c. (c) If a 71 and m6n, then am 6an. (d) If 0 6a 61 and m6n, then am 7an. Exponential Function If a 70 and a≠1, then the exponential function with base a is ƒ1x2 =ax. Characteristics of the Graph of ƒ1x2 =ax 1. The points A -1, 1 aB, 10, 12, and 11, a2 are on the graph. 2. If a 71, then ƒ is an increasing function. If 0 6a 61, then ƒ is a decreasing function. 3. The x-axis is a horizontal asymptote. 4. The domain is 1-∞, ∞2, and the range is 10, ∞2. 4.3 Logarithmic Functions Logarithm For all real numbers y and all positive numbers a and x, where a≠1, y =log a x is equivalent to x =a y. Logarithmic Function If a 70, a≠1, and x 70, then the logarithmic function with base a is ƒ1x2 =log a x. Characteristics of the Graph of ƒ1x2 =log a x 1. The points A1 a , -1B, 11, 02, and 1a, 12 are on the graph. 2. If a 71, then ƒ is an increasing function. If 0 6a 61, then ƒ is a decreasing function. 3. The y-axis is a vertical asymptote. 4. The domain is 10, ∞2, and the range is 1-∞, ∞2. Properties of Logarithms For x 70, y 70, a 70, a≠1, and any real number r, the following properties hold true. log a xy =log a x +log a y Product property log a x y =log a x −log a y Quotient property log a x r =r log a x Power property log a 1 =0 Logarithm of 1 log a a =1 Base a logarithm of a Theorem on Inverses For a 70 and a≠1, the following properties hold true. alog a x =x 1x +02 and log a ax =x log3 81 = 4 is equivalent to 34 = 81. Graph ƒ1x2 = log3 x. Give the domain and range. x y 1 31 1 0 3 1 0 1 3 x –2 1 y f(x) = log3 x (3, 1) (1, 0) Q , –1R 1 3 The domain is 10, ∞2, and the range is 1-∞, ∞2. Use the properties of logarithms to rewrite each expression. log2 13 # 52 = log2 3 + log2 5 log2 3 5 = log2 3 - log2 5 log6 35 = 5 log 6 3 log10 1 = 0 log10 10 = 1 2log2 5 = 5 and log 2 25 = 5
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