College Algebra (6th Edition)

Published by Pearson
ISBN 10: 0-32178-228-3
ISBN 13: 978-0-32178-228-1

Chapter 4 - Exponential and Logarithmic Functions - Exercise Set 4.3: 27

Answer

$2log_{b}x+log_{b}y-2log_{b}z$

Work Step by Step

Based on the quotient rule of logarithms, we know that $log_{b}(\frac{M}{N})=log_{b}M-log_{b}N$ (where $b$, $M$, and $N$ are positive real numbers and $b\ne1$). Therefore, $log_{b}(\frac{x^{2}y}{z^{2}})=log_{b}x^{2}y-log_{b}z^{2}$. Based on the product rule of logarithms, we know that $log_{b}(MN)=log_{b}M+log_{b}N$ (for $M\gt0$ and $N\gt0$). Therefore, $log_{b}x^{2}y-log_{b}z^{2}=log_{b}x^{2}+log_{b}y-log_{b}z^{2}$. According to the power rule of logarithms, we know that $log_{b}M^{p}=plog_{b}M$ (when $b$ and $M$ are positive real numbers, $b\ne1$, and $p$ is any real number). Therefore, $log_{b}x^{2}+log_{b}y-log_{b}z^{2}=2log_{b}x+log_{b}y-2log_{b}z$.
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