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
Step 1: Identify the equation and determine the domain. The equation shown in the image is (x-1) ÷ (x-1) = 2 (x-2). For the logarithms to be defined, their arguments must be positive: • For (x-1), we need x-1 > 0 x > 1. • For (x-2), we need x-2 > 0 x > 2. Additionally, the denominator (x-1) cannot be zero. This means x-1 ≠ 1, so x ≠ 2. Combining these conditions, the valid domain for x is x > 2. Step 2: Simplify the left side of the equation. Since x > 2, (x-1) is defined and non-zero. Any non-zero quantity divided by itself is 1. ((x-1))/((x-1)) = 1 So the equation simplifies to: 1 = 2 (x-2) Step 3: Isolate the logarithm term. Divide both sides by 2: (x-2) = (1)/(2) Step 4: Convert the logarithmic equation to exponential form. Assuming the base of the logarithm is 10 (common logarithm, as no base is specified): x-2 = 10^1/2 x-2 = sqrt(10) Step 5: Solve for x. Add 2 to both sides: x = 2 + sqrt(10) Step 6: Verify the solution against the domain. The approximate value of sqrt(10) is 3.162. So, x ≈ 2 + 3.162 = 5.162. This value satisfies the domain condition x > 2. Therefore, the solution is valid. The final answer is x = 2 + sqrt(10). 3 done, 2 left today. You're making progress.