Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 12 - Exponential Functions and Logarithmic Functions - 12.4 Properties of Logarithmic Functions - 12.4 Exercise Set: 44

Answer

$2\log_c x-\dfrac{3}{2}\log_c y-\log_cz$

Work Step by Step

Using the properties of logarithms, the given expression, $ \log_c \sqrt{\dfrac{x^4}{y^3z^2}} ,$ is equivalent to \begin{array}{l}\require{cancel} \log_c \left( \dfrac{x^4}{y^3z^2} \right)^{1/2} \\\\= \dfrac{1}{2}\log_c \dfrac{x^4}{y^3z^2} \\\\= \dfrac{1}{2}\left[ \log_c x^4-\log_c (y^3z^2) \right] \\\\= \dfrac{1}{2}\left[ \log_c x^4-\left( \log_c y^3+\log_cz^2 \right) \right] \\\\= \dfrac{1}{2}\left[ \log_c x^4-\log_c y^3-\log_cz^2 \right] \\\\= \dfrac{1}{2}\left[ 4\log_c x-3\log_c y-2\log_cz \right] \\\\= \dfrac{1}{2}(4\log_c x)-\dfrac{1}{2}(3\log_c y)-\dfrac{1}{2}(2\log_cz) \\\\= \dfrac{4}{2}\log_c x-\dfrac{3}{2}\log_c y-\dfrac{2}{2}\log_cz \\\\= 2\log_c x-\dfrac{3}{2}\log_c y-\log_cz .\end{array}
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