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) $4 \log _ { a } x + \frac { 1 } { 3 } \log _ { a } ( x + 5 ) - 2 \log _ { a } ( x - 2 )$
B) $4 \log _ { a } x - 3 \log _ { a } ( x + 5 ) - 2 \log _ { a } ( x - 2 )$
C) $\log _ { a } x ^ { 4 } + \log _ { a } ( x + 5 ) ^ { - 3 } - \log _ { a } ( x - 2 ) ^ { 2 }$
D) $\log _ { a } x ^ { 4 } + \log _ { a } ( x + 5 ) ^ { 1 / 3 } - \log _ { a } ( x - 2 ) ^ { 2 }$

Question

Quizplus · Accepted Answer

Using the properties of logarithms, the expression can be expanded as follows: The quotient rule of logarithms allows the division inside the log to be written as a difference of logs. The product inside the log can be written as a sum of logs. The power rule allows the exponents to be brought in front of the log. Therefore, the correct expansion is $4 \log _ { a } x + \frac { 1 } { 3 } \log _ { a } ( x + 5 ) - 2 \log _ { a } ( x - 2 )$.

Use Properties of Logarithms to Expand the Logarithmic Expression as Much