Solve for $x$.
\[
\log _{5}(x+2)=1-\log _{5}(x+6)
\]
If there is more than one solution, separate them with commas.
Answer
\(\boxed{\text{Therefore, there are no valid solutions to the equation.}}\)
Step 1 ：First, simplify the equation using the properties of logarithms. The equation \(\log _{5}(x+2)=1-\log _{5}(x+6)\) can be rewritten as \(\log _{5}\left(\frac{x+2}{x+6}\right)=1\).
Step 2 ：Next, convert the logarithmic equation into an exponential equation. This gives \(5^1 = \frac{x+2}{x+6}\).
Step 3 ：Solving this equation for \(x\) gives \(x = -7\).
Step 4 ：However, this solution is not valid because it would result in a negative argument for the logarithm in the original equation, which is undefined.
Step 5 ：\(\boxed{\text{Therefore, there are no valid solutions to the equation.}}\)
