Which expression is equivalent to log sub 12 the fraction with numerator x to the 4th power square root of x cubed minus 2 end root and denominator open paren x plus 1 close paren to the 5th power ?

Use the quotient rule to subtract the logarithm of the denominator, and the product rule to split the numerator: log₁₂(x⁴) + log₁₂(√(x³ − 2)) − log₁₂((x + 1)⁵). Then use log₁₂(aⁿ) = n log₁₂(a), with √(x³ − 2) = (x³ − 2)^(1/2). This gives 4 log₁₂ x + ½ log₁₂(x³ − 2) − 5 log₁₂(x + 1), which is choice D.

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