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.