Intermediate Algebra: Connecting Concepts through Application

Published by Brooks Cole
ISBN 10: 0-53449-636-9
ISBN 13: 978-0-53449-636-4

Chapter 6 - Logarithmic Functions - 6.6 Solving Logarithmic Equations - 6.6 Exercises: 55

Answer

$x = -13.5, -5.5$

Work Step by Step

$\log_4 (-2x +5) + \log_4 (x+21.5) = 4$ $\log_4 (-2x+5)(x+21.5) = 4$ $4^{4} = (-2x+5)(x+21.5)$ $-2x(x+21.5)+5(x+21.5)= 256$ $-2x^{2} - 43x + 5x + 107.5 = 256$ $-2x^{2} - 38x + 107.5 - 256 = 0$ $-2x^{2} - 38x - 148.5 = 0$ $x = \frac{-b±\sqrt {b^{2}-4ac}}{2a}$ $x = \frac{-(-38)±\sqrt {(-38)^{2}-4(-2)(-148.5)}}{2(-2)}$ $x = \frac{38±\sqrt {1444-1188}}{-4}$ $x = \frac{38±\sqrt {256}}{-4}$ $x = \frac{38±16}{-4}$ $x = -13.5, -5.5$ Check: When $x = -5.5$ $\log_4 (-2(-5.5) +5) + \log_4 ((-5.5)+21.5) \overset{?}{=} 4$ $\log_4 (11 +5) + \log_4 (16) \overset{?}{=} 4$ $\log_4 (16) + \log_4 (16) \overset{?}{=} 4$ $2 + 2 \overset{?}{=} 4$ $4 = 4$ When $x = -13.5$ $\log_4 (-2(-13.5) +5) + \log_4 ((-13.5)+21.5) \overset{?}{=} 4$ $\log_4 (27 +5) + \log_4 (8) \overset{?}{=} 4$ $\log_4 (32) + \log_4 (8) \overset{?}{=} 4$ $2.5 + 1.5 \overset{?}{=} 4$ $4 = 4$
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