Question

Expand the logarithmic expression:

log((x³)² / ((x-2)(x²+5)⁴))
(a) 2log(x³) - log((x-2)(x²+5)⁴)
(b) log(x⁶) - log(x-2) - 4log(x²+5)
(c) 2log(x³) - log(x-2) - 16log(x²+5)
(d) log(x⁶) - log((x²+5)⁴)

Answer 1

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.