College Algebra, Chapter 5, 5.4, Section 5.4, Problem 50

Solve the Logarithmic Equation $\log_3 ( x + 15 ) - \log_3 ( x - 1) = 2$ for $x$.

$
\begin{equation}
\begin{aligned}
\log_3 ( x + 15 ) - \log_3 ( x - 1) &= 2\\
\\
\log_3 \frac{x+15}{x-1} &= 2 && \text{Laws of Logarithms } \log_a \frac{A}{B} = \log_a A - \log_a B\\
\\
3^{\log_3 \frac{x+15}{x-1}} &= 3^2 && \text{Raise 3 to each side}\\
\\
\frac{x+15}{x-1} &= 9 && \text{Property of log}\\
\\
x + 15 &= 9 (x - 1) && \text{Multiply each side by } x -1\\
\\
x + 15 &= 9x - 1 && \text{Distributive property}\\
\\
15 + 1 &= 9x - x && \text{Combine like terms}\\
\\
16 &= 8x && \text{Solve for } x\\
\\
x &= 2 && \text{Divide by 8}
\end{aligned}
\end{equation}
$