Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 3 - Review Exercises - Page 513: 53


$\displaystyle \frac{1}{3}\ln x-\frac{1}{3}$

Work Step by Step

Basic logarithmic properties:$ \left\{\begin{array}{l} \log_{b}1=0\\ \log_{b}b=1\\ \log_{b}b^{x}=x\\ b^{\log_{b}}x=x \end{array}\right.$ Rules: The Product Rule: $\log_{b}(MN)=\log_{b}\mathrm{M}+\log_{b}\mathrm{N}$ The Quotient Rule: $\displaystyle \log_{b}(\frac{M}{N})=\log_{b}\mathrm{M}-\log_{b}\mathrm{N}$ The Power Rule: $\log_{b}(M^{p})=p\cdot\log_{b}\mathrm{M}$ --- $\displaystyle \ln\sqrt[3]{\frac{x}{e}}=\ln(\frac{x}{e})^{1/3}=\qquad $... apply $:$ Power Rule $=\displaystyle \frac{1}{3}\ln(\frac{x}{e})\qquad $... apply $:$ Quotient Rule $=\displaystyle \frac{1}{3}[\ln x-\ln e]\qquad $... apply $: \log_{b}b=1, (\ln e=\log_{e}e)$ $=\displaystyle \frac{1}{3}[\ln x-1] $ $=\displaystyle \frac{1}{3}\ln x-\frac{1}{3}$
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