Logarithms Practice Test 2

Concept:Use the identity $(a-b)^3 = a^3 - b^3 - 3ab(a-b)$ and the log power rule $\log x^n = n\log x$.Explanation:Assume the intended expression is $\log[(a^3-b^3)-3ab(a-b)] - 3\log(a-b)$.First, simplify the term inside the first logarithm.Expand $(a-b)^3$ to get $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$.Rewrite this as $a^3 - b^3 - 3ab(a-b)$.Thus, $a^3 - b^3 - 3ab(a-b) = (a-b)^3$.So the expression becomes $\log((a-b)^3) - 3\log(a-b)$.Recall that $\log(x^n) = n\log x$.Apply the power rule: $\log((a-b)^3) = 3\log(a-b)$.Substitute this back: $3\log(a-b) - 3\log(a-b) = 0$.Both logarithmic terms are identical, so they cancel completely.Therefore, the value is $0$.Answer:$0$