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: x + (2x - 5) = 3 Step 1: Apply the product rule of logarithms, a + b = (ab), to the left side of the equation. (x(2x - 5)) = 3 Step 2: Since A = B implies A = B, equate the arguments of the logarithms. x(2x - 5) = 3 Step 3: Expand the left side and rearrange the equation into a standard quadratic form ax^2 + bx + c = 0. 2x^2 - 5x = 3 2x^2 - 5x - 3 = 0 Step 4: Solve the quadratic equation. We can use factoring or the quadratic formula. Let's factor. We need two numbers that multiply to 2 × (-3) = -6 and add to -5. These numbers are -6 and 1. 2x^2 - 6x + x - 3 = 0 2x(x - 3) + 1(x - 3) = 0 (2x + 1)(x - 3) = 0 This gives two possible solutions for x: 2x + 1 = 0 2x = -1 x = -(1)/(2) x - 3 = 0 x = 3 Step 5: Check the validity of the solutions. The argument of a logarithm must be positive. For x = -(1)/(2): The term x becomes (-(1)/(2)), which is undefined for real numbers. So, x = -(1)/(2) is not a valid solution. For x = 3: The term x becomes 3, which is valid. The term (2x - 5) becomes (2(3) - 5) = (6 - 5) = 1, which is valid. Therefore, x = 3 is the only valid solution. The value of x is 3. What's next? 📸