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: 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, $log\sqrt[3] \frac{x}{y}=log_{10}\sqrt[3] \frac{x}{y}=log_{10}\frac{\sqrt[3] x}{\sqrt[3] y}$. 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, $log_{10}\frac{\sqrt[3] x}{\sqrt[3] y}=log_{10}\sqrt[3] x-log_{10}\sqrt[3] y=log_{10}x^{\frac{1}{3}}-log_{10}y^{\frac{1}{3}}$. 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, $log_{10}x^{\frac{1}{3}}-log_{10}y^{\frac{1}{3}}=\frac{1}{3}log_{10}x-\frac{1}{3}log_{10}y$.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.