College Algebra (11th Edition)

Published by Pearson

Chapter 4 - Section 4.3 - Logarithmic Functions - 4.3 Exercises - Page 424: 89

Answer

$0.3522$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Use the Laws of Logarithms to find an equivalent expression for the given expression, $\log_{10} \dfrac{9}{4} ,$ which uses the given logarithmic values, $\log_{10} 2= 0.3010$ and/or $\log_{10} 3= 0.4771 .$ $\bf{\text{Solution Details:}}$ The given expression can be expressed as \begin{array}{l}\require{cancel} \log_{10} \dfrac{3^2}{2^2} .\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 \begin{array}{l}\require{cancel} \log_{10} 3^2-\log_{10}2^2 .\end{array} Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the expression above is equivalent \begin{array}{l}\require{cancel} 2\log_{10} 3-2\log_{10}2 .\end{array} Substituting the known values of the logarithmic expressions results to \begin{array}{l}\require{cancel} 2(0.4771)-2(0.3010) \\\\= 0.9542-0.6020 \\\\= 0.3522 .\end{array}

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