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
You're on a roll — let's solve this logarithmic equation. Given the equation: _2 x - _2(x - 2) = 3 Step 1: Apply the quotient rule of logarithms, _b A - _b B = _b ((A)/(B)), to the left side of the equation. _2 ((x)/(x - 2)) = 3 Step 2: Convert the logarithmic equation into its equivalent exponential form. The definition of a logarithm states that _b a = c is equivalent to b^c = a. Applying this to our equation, where b=2, a=(x)/(x-2), and c=3: 2^3 = (x)/(x - 2) Step 3: Simplify the left side and solve for x. 8 = (x)/(x - 2) Multiply both sides by (x - 2): 8(x - 2) = x 8x - 16 = x Subtract x from both sides: 7x - 16 = 0 Add 16 to both sides: 7x = 16 Divide by 7: x = (16)/(7) Step 4: Check the validity of the solution. The arguments of the original logarithms must be positive. For _2 x, we need x > 0. Our solution x = (16)/(7) satisfies this condition. For _2(x - 2), we need x - 2 > 0, which means x > 2. Our solution x = (16)/(7) ≈ 2.286, which is greater than 2. Both conditions are satisfied, so the solution is valid. The value of x is (16)/(7). Drop the next question!