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


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