College Algebra, Chapter 5, 5.3, Section 5.3, Problem 44

Expand the expression $\displaystyle \log \left( \frac{10}{x (x^2 + 1)(x^4 + 2)} \right) $, using Laws of Logarithm


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\begin{equation}
\begin{aligned}

\log \left( \frac{10}{x (x^2 + 1)(x^4 + 2)} \right) =& \log 10 - \log x (x^2 + 1)(x^4 + 1)
&& \text{Law of Logarithm } \log_a \left( \frac{A}{B} \right) = \log_a A - \log_a B
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\log \left( \frac{10}{x (x^2 + 1)(x^4 + 2)} \right) =& \log 10 - \left[ \log x + \log (x^2 + 1) + \log (x^4 + 1) \right]
&& \text{Law of Logarithm } \log_a (AB) = \log_a A + \log_a B
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\log \left( \frac{10}{x (x^2 + 1)(x^4 + 2)} \right) =& \log 10 - \log x - \log (x^2 + 1) - \log (x^4 + 1)
&& \text{Distributive Property}

\end{aligned}
\end{equation}
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