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 — yes, the method shown in the image is correct, and the steps lead to the right answer. There's just a minor notation point in the third line. Let's go through the steps clearly: Given the equation: _6 (2x - 4) + _6 (4) = _6 40 Step 1: Apply the product rule of logarithms, _b A + _b B = _b (AB), to the left side of the equation. _6 ( (2x - 4) · 4 ) = _6 40 Step 2: Distribute the 4 into the term (2x - 4). _6 (8x - 16) = _6 40 (Note: The image showed _6 8x - 16 = _6 40, which could be misinterpreted as _6(8x) minus 16. The correct interpretation, as used in the subsequent steps, is that 8x-16 is the argument of the logarithm, so it should be enclosed in parentheses.) Step 3: Since the logarithms on both sides have the same base and are equal, their arguments must be equal. 8x - 16 = 40 Step 4: Add 16 to both sides of the equation. 8x = 40 + 16 8x = 56 Step 5: Divide both sides by 8 to solve for x. x = (56)/(8) x = 7 Step 6: Check the validity of the solution. The argument of a logarithm must be positive. For x = 7: 2x - 4 = 2(7) - 4 = 14 - 4 = 10. Since 10 > 0, the logarithm _6(10) is defined. The other arguments (4 and 40) are already positive. Thus, x=7 is a valid solution. The final answer is 7. What's next? 📸