Single Variable Calculus, Chapter 7, 7.2-2, Section 7.2-2, Problem 8

Express the quantity $\displaystyle \ln (a + b) + \ln(a - b) - 2 \ln c$ as a single at logarithm.


$\begin{equation} \begin{aligned} \ln (a + b) + \ln (a - b) - 2 \ln c =& \ln [(a + b)(a - b)] - 2 \ln c && \text{(recall that } \ln x + \ln y = \ln (xy)) \\ \\ \ln (a + b) + \ln (a - b) - 2 \ln c =& \ln [(a + b)(a - b)] - \ln (c)^2 && \text{(recall that } k \ln x = \ln (x)^k) \\ \\ \ln (a + b) + \ln (a - b) - 2 \ln c =& \ln \left[ \frac{(a + b)(a - b)}{e^2} \right] && \text{(recall that } \ln x - \ln y = \ln \frac{x}{y} \\ \\ \ln (a + b) + \ln (a - b) - 2 \ln c =& \ln \left(\frac{a^2 - b^2}{c^2} \right) && \end{aligned} \end{equation}$