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: Apply the power rule of logarithms, a _b x = _b (x^a), to each term. 2_3 9 = _3 (9^2) = _3 81 (1)/(2)_3 144 = _3 (144^1/2) = _3 sqrt(144) = _3 12 The expression becomes: _3 81 - _3 12 + _3 972 Step 2: Apply the quotient and product rules of logarithms, _b x - _b y = _b ((x)/(y)) and _b x + _b y = _b (xy). Combine the terms into a single logarithm: _3 81 - _3 12 + _3 972 = _3 ((81)/(12) × 972) _3 ((81 × 972)/(12)) Step 3: Simplify the argument of the logarithm. Divide 972 by 12: 972 ÷ 12 = 81 Substitute this back into the argument: (81 × 972)/(12) = 81 × 81 = 81^2 Step 4: Express the argument as a power of the base 3. We know that 81 = 3^4. So, 81^2 = (3^4)^2 = 3^4 × 2 = 3^8. Step 5: Substitute this back into the logarithm. The expression becomes: _3 (3^8) Step 6: Use the property _b (b^n) = n. _3 (3^8) = 8 The final answer is 8. Send me the next one 📸