College Algebra, Chapter 5, 5.3, Section 5.3, Problem 10

Evaluate the expression $\displaystyle \log \frac{1}{\sqrt{1000}}$


$\begin{equation} \begin{aligned} \log \frac{1}{\sqrt{1000}} =& \log 1 - \log \sqrt{1000} && \text{Law of Logarithms } \log_a \left( \frac{A}{B} \right) = \log_a A - \log_a B \\ \\ \log \frac{1}{\sqrt{1000}} =& 0 - \log \sqrt{1000} && \text{Properties of Logarithms } \log_a 1 = 0 \\ \\ \log \frac{1}{\sqrt{1000}} =& \frac{-3}{2} && \text{Because } 10^{\frac{-3}{2}} = \frac{1}{\sqrt{1000}} \end{aligned} \end{equation}$