Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 4 - Review - Exercises - Page 389: 55


$$\log (\frac{x^2-4}{\sqrt {x^2+4}})$$

Work Step by Step

$Combine$ $into$ $a$ $single$ $logarithm:$ $\log (x-2)$ + $\log (x+2)$ - $\frac{1}{2}\log (x^2+4)$ Use the Third Law of Logarithms for $\frac{1}{2}\log (x^2+4)$ $\frac{1}{2}\log (x^2+4)$ = $\log (x^2+4)^{\frac{1}{2}}$ $\log (x-2)$ + $\log (x+2)$ - $\log (x^2+4)^{\frac{1}{2}}$ Use the First Law of Logarithms for $\log (x-2)$ + $\log (x+2)$ $\log (x-2)$ + $\log (x+2)$ = $\log ((x-2)(x+2))$ $\log ((x-2)(x+2))$ - $\log (x^2+4)^{\frac{1}{2}}$ Use the Second Law of Logarithms $\log ((x-2)(x+2))$ - $\log (x^2+4)^{\frac{1}{2}}$ = $\log (\frac{(x-2)(x+2)}{(x^2+4)^{\frac{1}{2}}})$ $$\log (\frac{(x-2)(x+2)}{(x^2+4)^{\frac{1}{2}}})$$ Simplify $$\log (\frac{x^2-4}{\sqrt {x^2+4}})$$
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