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: 63

Answer

$log(x^{\frac{1}{2}}y^{\frac{1}{2}})$

Work Step by Step

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, $\frac{1}{2}(log(x)+log(y))=\frac{1}{2}log(xy)$. 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, $ \frac{1}{2}log(xy)=log(x^{\frac{1}{2}}y^{\frac{1}{2}})$. In this case, the given logarithm is a common logarithm with an understood base of 10.
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