Answer
$x=1005$
Work Step by Step
Using the properties of logarithms, the given equation, $
\log(x^2-25)-\log(x+5)=3
,$ is equivalent to
\begin{align*}
\log\dfrac{x^2-25}{x+5}=3
&(\log_b x-\log_b y=\log_b\dfrac{x}{y})
\\\\
\log_{10}\dfrac{x^2-25}{x+5}=3
.\end{align*}
Since $y=b^x$ implies $\log_b y=x,$ the equation above is equivalent to
\begin{align*}\require{cancel}
10^3&=\dfrac{x^2-25}{x+5}
.\end{align*}
Cancelling the common factor and using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
10^3&=\dfrac{\cancel{(x+5)}(x-5)}{\cancel{x+5}}
&(a^2-b^2=(a+b)(a-b))
\\\\
1000&=x-5
\\
1000+5&=x-5+5
\\
1005&=x
\\
x&=1005
.\end{align*}
Hence, the solution to the equation $\log(x^2-25)-\log(x+5)=3$ is $x=1005$.
