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
Welcome back Jacinta — missed you this week. Here are the solutions to your problems. Problem 1: Evaluate using logarithm tables Let Y = sqrt((3.894 × 0.0645)/(32.52)). Taking the logarithm base 10 on both sides: Y = ( ((3.894 × 0.0645)/(32.52))^1/2 ) Y = (1)/(2) ((3.894 × 0.0645)/(32.52)) Using logarithm properties, (A/B) = A - B and (A × B) = A + B: Y = (1)/(2) [(3.894) + (0.0645) - (32.52)] Step 1: Find the logarithms of the individual numbers. Using logarithm tables: • (3.894) = 0.5904 • (0.0645) = 2.8096 (which is -2 + 0.8096) • (32.52) = 1.5121 Step 2: Substitute these values into the equation for Y. Y = (1)/(2) [0.5904 + (2.8096) - 1.5121] First, calculate the sum of the logarithms in the numerator: (3.894 × 0.0645) = 0.5904 + 2.8096 = 0.5904 + (-2 + 0.8096) = (0.5904 + 0.8096) - 2 = 1.4000 - 2 = 1.4000 Next, subtract the logarithm of the denominator: ((3.894 × 0.0645)/(32.52)) = 1.4000 - 1.5121 = (-1 + 0.4000) - 1.5121 = -1 - 1.5121 + 0.4000 = -2.5121 + 0.4000 = -2.1121 Step 3: Multiply by (1)/(2). Y = (1)/(2) (-2.1121) = -1.05605 Step 4: Convert Y to characteristic-mantissa form (where the mantissa is positive). -1.05605 = -2 + (2 - 1.05605) = -2 + 0.94395 So, Y = 2.94395. Step 5: Find the antilog of 2.94395. The mantissa is 0.94395. From antilog tables, the number corresponding to 0.94395 is approximately 8.789. The characteristic is 2, which means the decimal point is two places to the left of the first digit. Y = 0.08789 The value is 0.08789. Problem 2: Find the value of x in the equation below. 125^x × 5^3x - 3125 = 0 Step 1: Isolate the terms with x. 125^x × 5^3x = 3125 Step 2: Express all numbers as powers of the same base. In this case, base 5. We know that 125 = 5^3 and 3125 = 5^5. Substitute these into the equation: (5^3)^x × 5^3x = 5^5 Step 3: Apply the exponent rules (a^m)^n = a^mn and a^m × a^n = a^m+n. 5^3x × 5^3x = 5^5 5^3x + 3x = 5^5 5^6x = 5^5 Step 4: Equate the exponents since the bases are the same. 6x = 5 Step 5: Solve for x. x = (5)/(6) The value of x is (5)/(6). Send me the next one 📸