College Algebra, Chapter 5, 5.3, Section 5.3, Problem 48

Combine the expression $\displaystyle \log_5 (x^2 - 1) - \log_5 (x - 1)$, using the Laws of Logarithm


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\begin{equation}
\begin{aligned}

\log_5 (x^2 - 1) - \log_5 (x - 1) =& \log_5 \left( \frac{x^2 - 1}{x - 1} \right)
&& \text{Law of Logarithm } \log_a \left( \frac{A}{B} \right) = \log_a A - \log_a B
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\log_5 (x^2 - 1) - \log_5 (x - 1) =& \log_5 \left[ \frac{(x - 1)(x + 1)}{(x - 1)} \right]
&& \text{Factor } x^2 - 1
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\log_5 (x^2 - 1) - \log_5 (x - 1) =& \log_5 (x + 1)
&& \text{Cancel out like terms}

\end{aligned}
\end{equation}
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