Final answer:
The expression (((a)/(b))² * c³) converted into a logarithmic equation using logarithmic properties would result in the equation 2log(a) - 2log(b) + 3log(c). None of the provided options A, B, C, or D matches this result. It seems there is a typo in the provided options as none include a subtraction as would be necessary in the conversion.
Step-by-step explanation:
The task here is to rewrite the given expression (((a)/(b))² * c³) as a logarithmic equation. We know that the logarithm of a product is equal to the sum of the logarithms of the factors, and the logarithm of a quotient is equal to the difference between the logarithm of the numerator and the logarithm of the denominator. Additionally, the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number, as stated previously.
Applying these rules to our expression, we have:
log(((a/b)² * c³)) = log((a/b)²) + log(c³) = 2*log(a/b) + 3*log(c) = 2*(log(a) - log(b)) + 3*log(c)
This simplifies to:
2log(a) - 2log(b) + 3log(c)
Since none of the answer options include a subtraction, it appears there may have been a typo in the options provided. Based on the properties applied, none of the given options (A, B, C, D) accurately reflect the transformed logarithmic equation of the original expression.