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}
$