Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 7 - Exponential and Logarithmic Functions - 7-6 Natural Logarithms - Practice and Problem-Solving Exercises - Page 481: 15

Answer

$\ln{\left(\dfrac{x^{\frac{1}{3}}y^{\frac{1}{3}}}{z^4}\right)}$

Work Step by Step

Distribute $\frac{1}{3}$ to obtain: \begin{align*} \frac{1}{3}\ln{x}+\frac{1}{3}\ln{y}-4\ln{z} \end{align*} Recall: (1) $n\cdot \ln{a}=\ln{a^n}$ (2) $\ln{a}+\ln{b}=\ln{(ab)}$ (3) $\ln{a}-\ln{b} = \ln{\left(\frac{a}{b}\right)}$ Use rule (1) above to obtain: \begin{align*} \frac{1}{3}\ln{x}+\frac{1}{3}\ln{y}-4\ln{z}&=\ln{\left(x^{\frac{1}{3}}\right)}+\ln{\left(y^{\frac{1}{3}}\right)}-\ln{\left(z^4\right)} \end{align*} Use rule (2) above to obtain: \begin{align*} \ln{\left(x^{\frac{1}{3}}\right)}+\ln{\left(y^{\frac{1}{3}}\right)}-\ln{\left(z^4\right)}&=\ln{\left(x^{\frac{1}{3}}y^{\frac{1}{3}}\right)}-\ln{(z^4)} \end{align*} Use rule (3) above to obtain: \begin{align*} \ln{\left(x^{\frac{1}{3}}y^{\frac{1}{3}}\right)}-\ln{(z^4)}&=\ln{\left(\frac{x^{\frac{1}{3}}y^{\frac{1}{3}}}{z^4}\right)} \end{align*}
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