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


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