College Algebra (10th Edition)

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

Chapter 6 - Review Exercises - Page 499: 23



Work Step by Step

The given expression is equivalent to: \begin{align*} &=\log{\left(x^2(x^3+1)^{\frac{1}{2}}\right)}\\ \end{align*} Recall: (1) $\log_a{(mn)}=\log_a{m} + \log_a{n}$ (2) $\log_a{\left(\frac{m}{n}\right)}=\log_a{m} - \log_a{n}$ (3) $\log_a{a^m}=m\log_a{m}$ Use rule (1) above to obtain: \begin{align*} \log{\left(x^2(x^3+1)^{\frac{1}{2}}\right)}&=\log{\left(x^2\right)}+\log{(x^3+1)^{\frac{1}{2}}} \end{align*} Use rule (3) above to obtain: \begin{align*} \log{(x^2)}+\log{(x^3+1)^{\frac{1}{2}}}&=2\log{x}+\frac{1}{2}\log{(x^3+1)} \end{align*}
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