Precalculus: Mathematics for Calculus, 7th Edition

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

Chapter 4 - Section 4.4 - Laws of Logarithms - 4.4 exercises: 40

Answer

$\log_{a}(\dfrac{x^{2}}{yz^{3}})=2\log_{a}x-\log_{a}y-3\log_{a}z$

Work Step by Step

$\log_{a}(\dfrac{x^{2}}{yz^{3}})$ The logarithm of a division can be expanded as a substraction. Like this: $\log_{a}(\dfrac{x^{2}}{yz^{3}})=\log_{a}(x^{2})-\log_{a}(yz^{3})=...$ The logarithm of a product can be expanded as a sum. We can use this law to expand $\log_{a}(yz^{3})$: $...=\log_{a}(x^{2})-[\log_{a}y+\log_{a}(z^{3})]=...$ $...=\log_{a}(x^{2})-\log_{a}y-\log_{a}(z^{3})=...$ We can take the exponents present in $\log_{a}(x^{2})$ and $\log_{a}(z^{3})$ to the front of their respective logarithms to multiply: $...=2\log_{a}x-\log_{a}y-3\log_{a}z$
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