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 given expression can be expanded as follows: The quotient rule of logarithms allows the division inside the log to be written as a difference: $\log_a\left(\frac{x^4\sqrt[3]{x+5}}{(x-2)^2}\right) = \log_a(x^4\sqrt[3]{x+5}) - \log_a((x-2)^2)$. The product inside the log can be written as a sum using the product rule: $\log_a(x^4) + \log_a(\sqrt[3]{x+5})$. The power rule allows the exponent to be brought in front of the log: $4\log_a(x) + \frac{1}{3}\log_a(x+5)$. Combining these steps gives the final expanded form: $4\log_a(x) + \frac{1}{3}\log_a(x+5) - 2\log_a(x-2)$, which matches choice A.

Use Properties of Logarithms to Expand the Logarithmic Expression as Much