## College Algebra (6th Edition)

Published by Pearson

# Chapter 4 - Exponential and Logarithmic Functions - Exercise Set 4.3 - Page 477: 31

#### Answer

$\frac{1}{3}log_{10}x-\frac{1}{3}log_{10}y$

#### Work Step by Step

We know that $log(x)$ is a common logarithm with an understand base of 10. Therefore, $log\sqrt \frac{x}{y}=log_{10}\sqrt \frac{x}{y}=log_{10}\frac{\sqrt x}{\sqrt y}$. 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, $log_{10}\frac{\sqrt x}{\sqrt y}=log_{10}\sqrt x-log_{10}\sqrt y=log_{10}x^{\frac{1}{3}}-log_{10}y^{\frac{1}{3}}$. 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, $log_{10}x^{\frac{1}{3}}-log_{10}y^{\frac{1}{3}}=\frac{1}{3}log_{10}x-\frac{1}{3}log_{10}y$.

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