Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 12 - Exponential Functions and Logarithmic Functions - 12.6 Solving Exponential Equations and Logarithmic Equations - 12.6 Exercise Set - Page 825: 65

Answer

$x=4$

Work Step by Step

First, the solutions must satisfy $\left\{\begin{array}{l} x-2\gt 0\\ x\gt 0 \end{array}\right.\quad \Rightarrow x\gt 2\qquad (*)$ in order for the equation to be defined. LHS: Apply$ \quad\log_{a}(MN)=\log_{a}M+\log_{a}N$ RHS: Apply $\quad \log_{2}2^{3}=3$ $\log_{2}[ x(x-2)]=\log_{2}8$ ... apply the principle of logarithmic equality $x(x-2)=8$ $ x^{2}-2x-8=0\quad$to factor, find factors of $-8$ with sum $-2$ ... these are $-4$ and $+2$ $(x+2)(x-4)=0$ Possible solutions: $ x=-2\qquad$... does not satisfy (*), not a solution. $x=4\qquad $... satisfies (*), and is a valid solution.
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