## College Algebra (6th Edition)

Published by Pearson

# Chapter 4 - Exponential and Logarithmic Functions - Exercise Set 4.3 - Page 477: 62

#### Answer

$ln(\frac{x^{4}y^{7}}{z^{3}})$

#### Work Step by Step

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, $4ln(x)+7ln(y)-3ln(z)=ln(x^{4})+ln(y^{7})-ln(z^{3})$. 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, $ln(x^{4})+ln(y^{7})-ln(z^{3})=ln(x^{4}y^{7})-ln(z^{3})$. 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, $ln(x^{4}y^{7})-ln(z^{3})=ln(\frac{x^{4}y^{7}}{z^{3}})$. In this case, the given logarithm is a natural logarithm with an understood base of $e$.

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