Solve the following logarithmic equation. Use a calculator if appropriate. \[ \log _{12}(x-2)-\log _{12}(x+7)=\log _{12} 3 \]

Step 1 ：Given the logarithmic equation \(\log _{12}(x-2)-\log _{12}(x+7)=\log _{12} 3\)

Step 2 ：Using the properties of logarithms, the difference of two logarithms is the logarithm of the quotient. So, we can rewrite the left side of the equation as a single logarithm: \(\log _{12}\left(\frac{x-2}{x+7}\right)=\log _{12} 3\)

Step 3 ：Then, we can use the property that if \(\log_b(a) = \log_b(c)\), then \(a = c\) to solve for x. This gives us the equation \(\frac{x - 2}{x + 7} = 3\)

Step 4 ：Solving this equation gives us the solution \(x = -\frac{23}{2}\)

Step 5 ：However, we need to check if this solution is valid. A logarithm is only defined for positive numbers. So, we need to check if \(x-2\) and \(x+7\) are both positive for \(x = -\frac{23}{2}\)

Step 6 ：Substituting \(x = -\frac{23}{2}\) into \(x-2\) and \(x+7\) gives us \(-\frac{27}{2}\) and \(-\frac{9}{2}\) respectively

Step 7 ：Since both \(x-2\) and \(x+7\) are not positive for \(x = -\frac{23}{2}\), the solution is not valid

Step 8 ：\(\boxed{\text{The equation has no solution}}\)