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
To evaluate _10 1.125 without a calculator, we first convert 1.125 into a fraction and then use the given logarithm values and properties. Step 1: Convert the decimal to a fraction. 1.125 = (1125)/(1000) Step 2: Simplify the fraction. Divide the numerator and denominator by common factors. (1125)/(1000) = (225)/(200) = (45)/(40) = (9)/(8) Step 3: Rewrite the logarithm using the simplified fraction. _10 1.125 = _10 ((9)/(8)) Step 4: Apply the logarithm quotient rule, ((a)/(b)) = a - b. _10 ((9)/(8)) = _10 9 - _10 8 Step 5: Express 9 and 8 in terms of 2 and 3. 9 = 3^2 8 = 2^3 Substitute these into the expression: _10 (3^2) - _10 (2^3) Step 6: Apply the logarithm power rule, (a^b) = b a. 2 _10 3 - 3 _10 2 Step 7: Substitute the given values _10 2 = 0.3010 and _10 3 = 0.4771. 2(0.4771) - 3(0.3010) Step 8: Perform the multiplications. 0.9542 - 0.9030 Step 9: Perform the subtraction. 0.0512 Step 10: Round the result to 2 significant figures. The first non-zero digit is 5, and the second is 1. The digit after 1 is 2, which is less than 5, so we round down. 0.051 The final answer is 0.051. Send me the next one 📸