Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
-$\log _ { a } \left( \frac { x ^ { 4 } \sqrt [ 3 ] { x + 5 } } { ( x - 2 ) ^ { 2 } } \right)$
A) $\log _ { a } x ^ { 4 } + \log _ { a } ( x + 5 ) ^ { 1 / 3 } - \log _ { a } ( x - 2 ) ^ { 2 }$
B) $4 \log _ { a } x + \frac { 1 } { 3 } \log _ { a } ( x + 5 ) - 2 \log _ { a } ( x - 2 )$
C) $\log _ { a } x ^ { 4 } + \log _ { a } ( x + 5 ) ^ { - 3 } - \log _ { a } ( x - 2 ) ^ { 2 }$
D) $4 \log _ { a } x - 3 \log _ { a } ( x + 5 ) - 2 \log _ { a } ( x - 2 )$

Question

Quizplus · Accepted Answer

The correct expansion uses the properties of logarithms: the quotient rule $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$, the product rule $\log_b(xy) = \log_b(x) + \log_b(y)$, and the power rule $\log_b(x^c) = c\log_b(x)$. Applying these rules to the given expression results in $4 \log _ { a } x + \frac { 1 } { 3 } \log _ { a } ( x + 5 ) - 2 \log _ { a } ( x - 2 )$.

Use Properties of Logarithms to Expand the Logarithmic Expression as Much