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 logarithm property _b(x) - _b(y) = _b((x)/(y)). The given expression is _3(40.5) - _3(10.5). _3(40.5) - _3(10.5) = _3((40.5)/(10.5)) Step 2: Simplify the fraction inside the logarithm. To simplify (40.5)/(10.5), multiply the numerator and denominator by 10 to remove the decimals. (40.5)/(10.5) = (40.5 × 10)/(10.5 × 10) = (405)/(105) Divide both the numerator and the denominator by their greatest common divisor. Both are divisible by 5. (405 ÷ 5)/(105 ÷ 5) = (81)/(21) Both 81 and 21 are divisible by 3. (81 ÷ 3)/(21 ÷ 3) = (27)/(7) Step 3: Substitute the simplified fraction back into the logarithm. _3((27)/(7)) Step 4: Apply the logarithm property _b((x)/(y)) = _b(x) - _b(y). _3((27)/(7)) = _3(27) - _3(7) Step 5: Evaluate _3(27). Since 27 = 3^3, we have _3(27) = 3. Step 6: Substitute the value of _3(27) into the expression. 3 - _3(7) Assuming the notation _3^7 in the options means _3(7), the simplified expression matches option C. The final answer is 3 - _3^7.