Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 4 - Exponential and Logarithmic Functions - Section 4.6 Logarithmic and Exponential Equations - 4.6 Assess Your Understanding - Page 337: 18



Work Step by Step

The domain of the variable requires that $x>0$ and $x-21>0$ This means that $x>21$. Recall: $\log_a (MN) = \log_a M+\log_a N$ Thus, the given equation is equivalent to: $\log(x(x-21))= 2\\ \log(x^2-21x)=2$ Recall further that: $y = \log_a b \text{ is equivalent to } b=a^y$. Therefore, $x^2-21x = 10^2$ $x^2-21x = 100$ $x^2-21x-100=0$ By Factoring: $(x+4)(x-25)=0$ Using the Zero-Product Property: $x+4=0 \hspace{5pt} \hspace{15pt} \text{or} \hspace{15pt} x-25 =0\\ x=-4 \text{ or } x=25$ Since $x>21$, then $x$ cannoe be $-4$ Hence, $x= \boxed{25}$
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