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 — Step 1: Divide the entire equation by 2. (2 _2 x - 2 _2 (8 - 3x))/(2) = (1)/(2) _2 x - _2 (8 - 3x) = (1)/(2) Step 2: Apply the quotient rule of logarithms, _b M - _b N = _b ((M)/(N)). _2 ((x)/(8 - 3x)) = (1)/(2) Step 3: Convert the logarithmic equation to an exponential equation. If _b M = c, then b^c = M. (x)/(8 - 3x) = 2^(1)/(2) (x)/(8 - 3x) = sqrt(2) Step 4: Solve for x. Multiply both sides by (8 - 3x). x = sqrt(2)(8 - 3x) x = 8sqrt(2) - 3sqrt(2)x Move all terms containing x to one side. x + 3sqrt(2)x = 8sqrt(2) Factor out x. x(1 + 3sqrt(2)) = 8sqrt(2) Divide to isolate x. x = 8sqrt(2)1 + 3sqrt(2) Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 1 - 3sqrt(2). x = 8sqrt(2)1 + 3sqrt(2) × 1 - 3sqrt(2)1 - 3sqrt(2) x = 8sqrt(2)(1 - 3sqrt(2))1^2 - (3sqrt(2))^2 x = 8sqrt(2) - 8sqrt(2) · 3sqrt(2)1 - (9 · 2) x = 8sqrt(2) - 24 · 21 - 18 x = 8sqrt(2) - 48-17 x = 48 - 8sqrt(2)17 Step 5: Check for valid solutions. The arguments of the logarithms must be positive. 1. x > 0 2. 8 - 3x > 0 8 > 3x x < (8)/(3) Let's approximate the value of x: x ≈ (48 - 8 × 1.414)/(17) ≈ (48 - 11.312)/(17) ≈ (36.688)/(17) ≈ 2.158 This value satisfies both conditions (2.158 > 0 and 2.158 < (8)/(3) ≈ 2.667). The solution is: x = 48 - 8sqrt(2)17 That's 4 down, 1 left today. Keep it up!