Evaluate \( \frac{\log_{5}(0.04)}{\log_{3}18-\log_{3}2} \)

Answer Details

To evaluate the expression, we need to use the properties of logarithms. First, we can simplify the expression inside the parentheses of the logarithm: log5(0.04) = log5(4/100) = log5(4) - log5(100) = log5(2^2) - log5(10^2) = 2log5(2) - 2 Next, we can simplify the second part of the expression: log318 - log32 = log3(18/32) = log3(9/16) = log3(3^2) - log3(2^4) = 2log3(3) - 4 Substituting these simplified expressions back into the original expression, we get: 2log5(2) - 2 - (2log3(3) - 4) Simplifying further, we get: 2log5(2) - 2 - 2log3(3) + 4 Combining like terms, we get: 2log5(2) - 2log3(3) + 2 Now, we can plug in the values of log5(2) and log3(3) using a calculator: log5(2) ≈ 0.4307 and log3(3) = 1 Substituting these values, we get: 2(0.4307) - 2(1) + 2 ≈ 0.8614 - 2 + 2 ≈ -0.1386 Therefore, the answer is -1, since it is the only option that is negative.