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: Simplify the first term of the equation. We know that _b 1 = 0 for any valid base b. Therefore, 5 _x 1 = 5 × 0 = 0. Step 2: Rewrite the equation with the simplified term. The equation becomes: 0 - (2)/(_x y) + _x y = -1 -(2)/(_x y) + _x y = -1 Step 3: Introduce a substitution to simplify the equation. Let P = _x y. Substitute P into the equation: -(2)/(P) + P = -1 Step 4: Solve the equation for P. Multiply the entire equation by P (assuming P ≠ 0): -2 + P^2 = -P Rearrange the terms to form a quadratic equation: P^2 + P - 2 = 0 Factor the quadratic equation: (P+2)(P-1) = 0 This gives two possible values for P: P = -2 or P = 1 Step 5: Substitute back P = _x y and solve for x in each case. Case 1: P = -2 _x y = -2 By the definition of logarithms, x^-2 = y. (1)/(x^2) = y x^2 = (1)/(y) Taking the square root of both sides, and considering that the base of a logarithm must be positive (x > 0): x = sqrt((1)/(y)) x = (1)/(sqrt(y)) Case 2: P = 1 _x y = 1 By the definition of logarithms, x^1 = y. x = y The solutions for x are: x = (1)/(sqrt(y)) or x = y 3 done, 2 left today. You're making progress.