469 4.3 Logarithmic Functions (c) The graph of ƒ1x2 = log4 1x + 22 + 1 is obtained by shifting the graph of y = log4 x to the left 2 units and up 1 unit. The domain is found by solving x + 2 70, which yields 1-2, ∞2. The vertical asymptote has been shifted to the left 2 units as well, and it has equation x = -2. The range is unaffected by the vertical shift and remains 1-∞, ∞2. See Figure 32. S Now Try Exercises 51, 55, and 59. y x 5 22 2 0 4 22 x (21, 1) (2, 2) f(x) 5 log4 (x 1 2) 1 1 Figure 32 Properties of Logarithms The properties of logarithms enable us to change the form of logarithmic statements so that products can be converted to sums, quotients can be converted to differences, and powers can be converted to products. Properties of Logarithms For x 70, y 70, a 70, a≠1, and any real number r, the following properties hold true. Property Description Product Property log a xy =log a x +log a y The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. Quotient Property log a x y =log a x −log a y The logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numbers. Power Property log a xr =r log a x The logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number. Logarithm of 1 log a 1 =0 The base a logarithm of 1 is 0. Base a Logarithm of a log a a =1 The base a logarithm of a is 1. Proof To prove the product property, let m= loga x and n = loga y. loga x = m means am = x Write in exponential form. loga y = n means an = y NOTE If we are given a graph such as the one in Figure 31 and asked to find its equation, we could reason as follows: The point 11, 02 on the basic logarithmic graph has been shifted down 1 unit, and the point 13, 02 on the given graph is 1 unit lower than 13, 12, which is on the graph of y = log3 x. Thus, the equation will be y = 1log3 x2 - 1.
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