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, c _b M = _b (M^c), to each term. (1)/(2)_3 36 = _3 (36^1/2) = _3 sqrt(36) = _3 6 3_3 3 = _3 (3^3) = _3 27 Step 2: Substitute these simplified terms back into the original expression. _3 6 - _3 2 - _3 27 Step 3: Apply the quotient rule of logarithms, _b M - _b N = _b ((M)/(N)), from left to right. First, combine the first two terms: _3 6 - _3 2 = _3 ((6)/(2)) = _3 3 Step 4: Combine the result with the third term. _3 3 - _3 27 = _3 ((3)/(27)) Step 5: Simplify the fraction inside the logarithm. _3 ((1)/(9)) Step 6: Express (1)/(9) as a power of 3. Since 9 = 3^2, then (1)/(9) = 3^-2. _3 (3^-2) Step 7: Use the definition of logarithm, _b (b^x) = x. _3 (3^-2) = -2 The simplified expression is: -2 3 done, 2 left today. You're making progress.