College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 6 - Section 6.5 - Properties of Logarithms - 6.5 Assess Your Understanding: 36

Answer

$\dfrac{a-b}{4}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To write the given expression, $ \ln\sqrt[4]{\dfrac{2}{3}} ,$ in terms of $a$ and $b,$ where $a=\ln2$ and $b=\ln3,$ use the laws of logarithms and substitution. $\bf{\text{Solution Details:}}$ The given expression is equivalent to \begin{array}{l}\require{cancel} \ln\left( \dfrac{2}{3} \right)^{1/4} .\end{array} Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the expression above is equivalent to: \begin{array}{l}\require{cancel} \dfrac{1}{4}\ln\dfrac{2}{3} .\end{array} Using the Quotient Rule of Logarithms, which is given by $\log_b \dfrac{x}{y}=\log_bx-\log_by,$ the expression above is equivalent to: \begin{array}{l}\require{cancel} \dfrac{1}{4}(\ln2-\ln3) .\end{array} By substitution, since $a=\ln2$ and $b=\ln3,$ the expression above is equivalent to: \begin{array}{l}\require{cancel} \dfrac{1}{4}(a-b) \\\\= \dfrac{a-b}{4} .\end{array}
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