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
Here are the solutions to the problems. 1. Express as a single logarithm: 3 + 16 - 4 - 6 Step 1: Use the logarithm property a + b = (a · b) to combine the positive terms and the negative terms separately. ( 3 + 16) - ( 4 + 6) (3 × 16) - (4 × 6) 48 - 24 Step 2: Use the logarithm property a - b = ((a)/(b)) to combine the remaining terms. ((48)/(24)) Step 3: Simplify the fraction. 2 The expression as a single logarithm is 2. 2. Simplify: i. _3 63 Step 1: Find the prime factorization of 63. 63 = 9 × 7 = 3^2 × 7 Step 2: Apply the logarithm properties _b (xy) = _b x + _b y and _b x^n = n _b x. _3 63 = _3 (3^2 × 7) _3 (3^2) + _3 7 2 _3 3 + _3 7 Step 3: Since _3 3 = 1, substitute this value. 2(1) + _3 7 2 + _3 7 The simplified expression is 2 + _3 7. ii. _3 150 Step 1: Find the prime factorization of 150. 150 = 15 × 10 = (3 × 5) × (2 × 5) = 2 × 3 × 5^2 Step 2: Apply the logarithm properties _b (xyz) = _b x + _b y + _b z and _b x^n = n _b x. _3 150 = _3 (2 × 3 × 5^2) _3 2 + _3 3 + _3 (5^2) _3 2 + _3 3 + 2 _3 5 Step 3: Since _3 3 = 1, substitute this value. _3 2 + 1 + 2 _3 5 The simplified expression is 1 + _3 2 + 2 _3 5.