College Algebra, Chapter 5, 5.4, Section 5.4, Problem 52

Solve the Logarithmic Equation $\log x + \log (x - 3) = 1$ for $x$.

$
\begin{equation}
\begin{aligned}
\log x + \log (x - 3) &= 1\\
\\
\log x(x -3) &= 1 && \text{Laws of Logarithm } \log_a AB = \log_a A + \log_a B\\
\\
10^{\log x(x -3)} &= 10^1 && \text{Raise 10 to each side}\\
\\
x(x - 3) &= 10 && \text{Property of log}\\
\\
x^2 - 3x &= 10 && \text{Distributive property}\\
\\
x^2 - 3x - 10 & = 0 && \text{Subtract 10 }\\
\\
(x - 5)(x + 2) &= 0 && \text{Factor}
\end{aligned}
\end{equation}
$

Solve for $x$

$
\begin{equation}
\begin{aligned}
x -5 &= 0 &&\text{and}& x + 2 &= 0 \\
\\
x &= 5 &&& x &= -2
\end{aligned}
\end{equation}
$

The only solution in the given equation is $x = 5$, since $x = -2$ will give a negative value.