This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
frankwellmwiya67, let's knock this out. Step 1: Apply the logarithm product rule, a + b = (ab), to the left side of the equation. ((x-1)(x+1)) = 2 (x-2) (x^2 - 1) = 2 (x-2) Step 2: Apply the logarithm power rule, c a = (a^c), to the right side of the equation. (x^2 - 1) = ((x-2)^2) Step 3: Since the logarithms on both sides are equal and have the same base, their arguments must be equal. x^2 - 1 = (x-2)^2 Step 4: Expand the right side and solve the resulting algebraic equation for x. x^2 - 1 = x^2 - 4x + 4 Subtract x^2 from both sides: -1 = -4x + 4 Add 4x to both sides and add 1 to both sides: 4x = 4 + 1 4x = 5 x = (5)/(4) Step 5: Check the validity of the solution. For a logarithm (A) to be defined, its argument A must be positive (A > 0). For (x-1), we need x-1 > 0 x > 1. For (x+1), we need x+1 > 0 x > -1. For (x-2), we need x-2 > 0 x > 2. The most restrictive condition is x > 2. Our calculated value is x = (5)/(4) = 1.25. Since 1.25 is not greater than 2, this solution is extraneous and not valid for the original equation. Therefore, there is no real solution for x. No solution Send me the next one 📸