Expand the logarithmic expression. $\log \left( \frac { 4 x ^ { 3 } } { \sqrt { y } ( x - 2 ) ^ { 6 } } \right)$
A) $2 \log 2 + 3 \log x - \frac { 1 } { 2 } \log y - 6 \log ( x - 2 )$
B) $\log 2 + 3 \log x - 2 \log y - 6 \log ( x - 2 )$
C) $2 \log 2 + 3 \log x - \frac { 1 } { 2 } \log y + 6 \log ( x - 2 )$
D) $2 \log 2 - 3 \log x - \log y - \frac { 1 } { 2 } \log ( x - 6 )$
E) none

Question

Quizplus · Accepted Answer

Using the properties of logarithms, the expression can be expanded as follows: $\log \left( \frac { 4 x ^ { 3 } } { \sqrt { y } ( x - 2 ) ^ { 6 } } \right)$ can be broken down into the sum and difference of logarithms. The numerator becomes $2 \log 2 + 3 \log x$, and the denominator becomes $- \frac { 1 } { 2 } \log y - 6 \log ( x - 2 )$, combining these gives the correct answer.

Expand the Logarithmic Expression log⁡(4x3y(x−2)6)\log \left( \frac { 4 x ^ { 3 } } { \sqrt { y } ( x - 2 ) ^ { 6 } } \right)log(y​(x−2)64x3​)

Expand the Logarithmic Expression $\log \left( \frac { 4 x ^ { 3 } } { \sqrt { y } ( x - 2 ) ^ { 6 } } \right)$