College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 4, Exponential and Logarithmic Functions - Chapter 4 Review - Concept Check: 55

Answer

$\log{\left(\dfrac{x^2-4}{\sqrt{x^2+4}}\right)}$

Work Step by Step

RECALL: (1) $n \cdot \log_a{P}=\log_a{(P^n)}$ (2) $\log_a{P} + \log_a{Q}=\log_a{(PQ)}$ (3) $\log_a{P} - \log_a{Q}=\log_a{(\frac{P}{Q})}$ (4) $a^m \cdot a^n=a^{m+n}$ Use rule (1) above to obtain $\log{(x-2)}+\log{(x+2)}-\frac{1}{2}\log{(x^2+4)} \\=\log{(x-2)}+\log{(x+2)}-\log{(x^2+4)^{\frac{1}{2}}}$ Use rule (2) above to obtain: $=\log{[(x-2)(x+2)]}-\log{x^2+4)^{\frac{1}{2}}}$ Since $(a-b)(a+b)=a^2-b^2$, then the expression above simplifies to: $=\log{(x^2-4)} - \log{(x^2+4)^{\frac{1}{2}}}$ Use rule (3) above to obtain: $=\log{\left(\dfrac{x^2-4}{(x^2+4)^{\frac{1}{2}}}\right)}$ Use the rule $a^{\frac{1}{2}}=\sqrt{a}$ to obtain: $=\log{\left(\dfrac{x^2-4}{\sqrt{x^2+4}}\right)}$
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