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


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