Chapter 12 - Exponential Functions and Logarithmic Functions - 12.4 Properties of Logarithmic Functions - 12.4 Exercise Set - Page 810: 54

$\log_a \dfrac{\sqrt{a}}{x}$

Work Step by Step

Using the properties of logarithms, the given expression, $\log_a \dfrac{a}{\sqrt{x}}-\log_a \sqrt{ax} ,$ is equivalent to \begin{array}{l}\require{cancel} \log_a \dfrac{\dfrac{a}{\sqrt{x}}}{\sqrt{ax}} \\\\= \log_a \dfrac{a}{\sqrt{x}\cdot\sqrt{ax}} \\\\= \log_a \dfrac{a}{\sqrt{ax^2}} \\\\= \log_a \dfrac{a}{x\sqrt{a}} \\\\= \log_a \left( \dfrac{a}{x\sqrt{a}}\cdot\dfrac{\sqrt{a}}{\sqrt{a}} \right) \\\\= \log_a \dfrac{a\sqrt{a}}{x\cdot a} \\\\= \log_a \dfrac{\sqrt{a}}{x} .\end{array}

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