College Algebra, Chapter 5, 5.3, Section 5.3, Problem 46

Combine the expression $\displaystyle \log 12 + \frac{1}{2} \log 7 - \log 2$, using the Laws of Logarithm


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\begin{equation}
\begin{aligned}

\log 12 + \frac{1}{2} \log 7 - \log 2 =& \log 12 + \log 7^{\frac{1}{2}} - \log 2
&& \text{Law of Logarithm } \log_a (A^C) = C \log_a A
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\log 12 + \frac{1}{2} \log 7 - \log 2 =& \log \left( 12 \sqrt{7} \right) - \log 2
&& \text{Law of Logarithm } \log_a (AB) = \log_a A + \log_a B
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\log 12 + \frac{1}{2} \log 7 - \log 2 =& \log \left( \frac{12 \sqrt{7}}{2} \right)
&& \text{Law of Logarithm } \log_a \left( \frac{A}{B} \right) = \log_a A - \log_a B
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\log 12 + \frac{1}{2} \log 7 - \log 2 =& \log \left( 6 \sqrt{7} \right)
&& \text{Reduce to lowest term}

\end{aligned}
\end{equation}
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