Solve the logarithmic equation. Be sure to reject any value of $x$ that is not in the domain of the original logarithmic expressions. Give the exact answer. \[ \log _{5}(x+7)+\log _{5}(x+3)=1 \] Rewrite the given equation without logarithms. Do not solve for $x$.

Step 1 ：Rewrite the given equation without logarithms. The equation can be rewritten as \(\log _{5}((x+7)(x+3))=1\).

Step 2 ：Use the definition of logarithms to rewrite this equation without logarithms. The definition of logarithms states that if \(\log _{b}(a)=c\), then \(b^c=a\). So, we can rewrite our equation as \(5^1=(x+7)(x+3)\).

Step 3 ：The equation without logarithms is \(\boxed{5=(x+7)(x+3)}\)