Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 7 - Exponential and Logarithmic Functions - 7-5 Exponential and Logarithmic Equations - Practice and Problem-Solving Exercises - Page 476: 91



Work Step by Step

Note that: \begin{align*} \sqrt{9x} &= \sqrt{3^2x} \\&=3\sqrt{x} \\&=3\cdot x^{\frac{1}{2}} \end{align*} Thus, the given expression is equivalent to: $$\log_3{\left(3x^{\frac{1}{2}}\right)}$$ Recall: (1) Product Property of Logarithms: $\log_a{b}+\log_a{c}=\log_a{bc}$. (2) Power Property of Logarithms: $\log_a{b^n} = n\cdot \log_a{b}$ Use the Product Property to obtain: \begin{align*} \log_3{\left(3x^{\frac{1}{2}}\right)}&=\log_3{3} + \log_3{x^{\frac{1}{2}}}\\ \end{align*} Use the Power Property to obtain: \begin{align*} \log_3{3} + \log_3{x^{\frac{1}{2}}}&=\log_3{3} +\frac{1}{2}\log_3{x}\\ \end{align*} Note that $\log_3{3}=1$. Thus, \begin{align*} \log_3{3} +\frac{1}{2}\log_3{x}&=1+\frac{1}{2}\log_3{x}\\ \end{align*}
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