3. Solve algebraically: \[ \log _{7}(x+4)+\log _{7}(x-2)=1 \]

Step 1 ：\(\log _{7}(x+4)+\log _{7}(x-2)=1\)

Step 2 ：Combine the two logarithms using the logarithmic property: \(\log(a) + \log(b) = \log(a * b)\)

Step 3 ：\(\log _{7}((x+4)(x-2))=1\)

Step 4 ：Use the property \(\log_{a}(b)=c\) to rewrite the equation as \(7^1 = (x+4)(x-2)\)

Step 5 ：\(7 = (x+4)(x-2)\)

Step 6 ：Solve the quadratic equation: \((x+4)(x-2) = 7\)

Step 7 ：\(x^2 + 2x - 8 = 0\)

Step 8 ：Factor the quadratic equation: \((x-2)(x+4) = 0\)

Step 9 ：Find the solutions: \(x = -5\) and \(x = 3\)

Step 10 ：\(\boxed{x = -5, x = 3}\)