College Algebra, Chapter 5, 5.3, Section 5.3, Problem 42

Expand the expression $\displaystyle \log \sqrt{x \sqrt{y \sqrt{z}}}$, using Laws of Logarithm


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\begin{equation}
\begin{aligned}

\log \sqrt{x \sqrt{y \sqrt{z}}} =& \frac{1}{2} \log \left( x \sqrt{y \sqrt{z}} \right)
&& \text{Law of Logarithm } \log_a (A^C) = C \log_a A
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\log \sqrt{x \sqrt{y \sqrt{z}}} =& \frac{1}{2} \left( \log x + \log \sqrt{y \sqrt{z}} \right)
&& \text{Law of Logarithm } \log_a (AB) = \log_a A + \log_a B
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\log \sqrt{x \sqrt{y \sqrt{z}}} =& \frac{1}{2} \left[ \log x + \frac{1}{2} \log (y \sqrt{z} ) \right]
&& \text{Law of Logarithm } \log_a (A^C) = C \log_a A
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\log \sqrt{x \sqrt{y \sqrt{z}}} =& \frac{1}{2} \left[ \log x + \frac{1}{2} \left( \log y + \log \sqrt{z} \right) \right]
&& \text{Law of Logarithm } \log_a (AB) = \log_a A + \log_a B
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\log \sqrt{x \sqrt{y \sqrt{z}}} =& \frac{1}{2} \left[ \log x + \frac{1}{2} \left( \log y + \frac{1}{2} \log z \right) \right]
&& \text{Law of Logarithm } \log_a (A^C) = C \log_a A
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\log \sqrt{x \sqrt{y \sqrt{z}}} =& \frac{1}{2} \log x + \frac{1}{4} \log y + \frac{1}{8} \log z
&& \text{Distributive Property}

\end{aligned}
\end{equation}
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