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's the solution for question 1b: Step 1: Combine the logarithms using the properties a + b = (ab) and a - b = ((a)/(b)). The given expression is 162 + 36 - 54. This can be written as: ((162 × 36)/(54)) Step 2: Simplify the expression inside the logarithm. First, simplify the fraction (36)/(54). Both numbers are divisible by 18: (36)/(54) = (36 ÷ 18)/(54 ÷ 18) = (2)/(3) Now substitute this back into the expression: (162 × (2)/(3)) Perform the multiplication: 162 × (2)/(3) = (162 × 2)/(3) = (324)/(3) = 108 So the expression simplifies to 108. Step 3: Express 108 as a product of its prime factors (2 and 3). 108 = 2 × 54 54 = 2 × 27 27 = 3 × 9 9 = 3 × 3 Therefore, 108 = 2 × 2 × 3 × 3 × 3 = 2^2 × 3^3. Step 4: Apply logarithm properties to expand 108. Using the property (ab) = a + b: 108 = (2^2 × 3^3) = (2^2) + (3^3) Using the property (a^b) = b a: (2^2) + (3^3) = 2 2 + 3 3 Step 5: Substitute the given values for 2 and 3. Given 3 = 0.4771 and 2 = 0.3010. 2 2 + 3 3 = 2(0.3010) + 3(0.4771) = 0.6020 + 1.4313 = 2.0333 The final answer is 2.0333. 3 done, 2 left today. You're making progress.