The expanded form of the logarithmic expression log((x³)² / ((x-2)(x²+5)⁴)) is 2×log(x³) - log(x-2) - 4×log(x²+5), aligning with option (c). This is obtained by applying the properties of logarithms that deal with division and exponents.
To expand the logarithmic expression log((x³)² / ((x-2)(x²+5)⁴)), we apply logarithmic properties. First, we use the property that log(a ÷ b) = log(a) - log(b). Second, we apply the property that log(a^n) = n × log(a), where 'n' is the exponent. This leads us to:
log((x³)²) - log((x-2)(x²+5)⁴)
Expanding further using the properties of exponents and logarithms:
2 × log(x³) - log(x-2) - log((x²+5)⁴)
Now we apply the property to the last term:
2 × log(x³) - log(x-2) - 4 × log(x²+5)
This matches option (c) in the expanded form of the original logarithmic expression.