Solve the logarithmic equation. \[ \log _{6}(x+8)+\log _{6}(x-8)=2 \]

Step 1 ：The given equation is a logarithmic equation with base 6. The equation is \(\log _{6}(x+8)+\log _{6}(x-8)=2\).

Step 2 ：Combine the two logarithms on the left side of the equation into one using the property of logarithms that states that the sum of two logarithms with the same base is the logarithm of the product of the numbers. This gives us \(\log _{6}((x+8)(x-8))=2\).

Step 3 ：Convert the logarithmic equation into an exponential equation to solve for x. This gives us \((x+8)(x-8)=6^2\).

Step 4 ：Solve the equation to find the possible values of x. The solutions are x = -10 and x = 10.

Step 5 ：Check these solutions in the original equation because logarithms are only defined for positive numbers. Therefore, x = -10 is not a valid solution because it would result in a negative number inside the logarithm.

Step 6 ：So, the only valid solution is x = 10.

Step 7 ：Final Answer: The solution to the equation is \(\boxed{10}\).