Rewrite as a single logarithm. Assume all variables represent positive real numbers.
-$4 \log ( x + 3 ) - 9 \log \left( x ^ { 2 } + 4 \right) + \frac { 1 } { 3 } \log y$
A) $- \frac { 14 } { 3 } \log \frac { ( x + 3 ) y } { x ^ { 2 } + 4 }$
B) $12 \log \frac { ( x + 3 ) \left( x ^ { 2 } + 4 \right) } { y }$
C) $\log \frac { ( x + 3 ) ^ { 4 } \sqrt [ 3 ] { y } } { \left( x ^ { 2 } + 4 \right) ^ { 9 } }$
D) $\log \frac { ( x + 3 ) ^ { 4 } \left( x ^ { 2 } + 4 \right) ^ { 9 } } { \sqrt [ 3 ] { y } }$

Question

Quizplus · Accepted Answer

Using logarithm properties, the expression can be simplified to a single logarithm by applying the power rule ($a \log b = \log b^a$) and the quotient rule ($\log a - \log b = \log \frac{a}{b}$). This results in option C, which correctly applies these rules to the given expression.

Rewrite as a Single Logarithm 4log⁡(x+3)−9log⁡(x2+4)+13log⁡y4 \log ( x + 3 ) - 9 \log \left( x ^ { 2 } + 4 \right) + \frac { 1 } { 3 } \log y4log(x+3)−9log(x2+4)+31​logy

Rewrite as a Single Logarithm $4 \log ( x + 3 ) - 9 \log \left( x ^ { 2 } + 4 \right) + \frac { 1 } { 3 } \log y$