# Chapter 12 - Exponential Functions and Logarithmic Functions - 12.4 Properties of Logarithmic Functions - 12.4 Exercise Set - Page 810: 45

$\displaystyle \frac{1}{3}[6\log_{a}x+3\log_{a}y-2-7\log_{a}z]$

#### Work Step by Step

$\displaystyle \log_{a}\sqrt[3]{\frac{x^{6}y^{3}}{a^{2}z^{7}}}=\log_{a}(\frac{x^{6}y^{3}}{a^{2}z^{7}})^{1/3}\qquad$ ... apply $\log_{a}M^{p}=p\cdot\log_{a}M$ $=\displaystyle \frac{1}{3}\log_{a}(\frac{x^{6}y^{3}}{a^{2}z^{7}})\qquad$ ... apply $\displaystyle \log_{a}\frac{M}{N}=\log_{a}M-\log_{a}N$ $=\displaystyle \frac{1}{3}[\log_{a}(x^{6}y^{3})-\log_{a}(a^{2}z^{7})]\qquad$ ... apply $\log_{a}(MN)=\log_{a}M+\log_{a}N$ $=\displaystyle \frac{1}{3}[\log_{a}x^{6}+\log_{a}y^{3}-(\log_{a}a^{2}+\log_{a}z^{7})]$ $=\displaystyle \frac{1}{3}[\log_{a}x^{6}+\log_{a}y^{3}-\log_{a}a^{2}-\log_{a}z^{7}]\qquad$ ... apply $\log_{a}M^{p}=p\cdot\log_{a}M$ $=\displaystyle \frac{1}{3}[6\log_{a}x+3\log_{a}y-2\log_{a}a-7\log_{a}z]\qquad$ ... apply $\log_{a}a=1$ $=\displaystyle \frac{1}{3}[6\log_{a}x+3\log_{a}y-2-7\log_{a}z]$

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