Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
-$\log _ { 3 } \left( \frac { 81 } { \sqrt { x - 1 } } \right)$
A) $4 \log _ { 3 } 3 - \frac { 1 } { 2 } \log _ { 3 } ( x - 1 )$
B) $\log _ { 3 } 81 - \log _ { 3 } \sqrt { x - 1 }$
C) $4 - \log _ { 3 } \sqrt { x - 1 }$
D) $4 - \frac { 1 } { 2 } \log _ { 3 } ( x - 1 )$

Question

Quizplus · Accepted Answer

The expression can be expanded using the properties of logarithms: $\log _ { 3 } \left( \frac { 81 } { \sqrt { x - 1 } } \right) = \log _ { 3 } 81 - \log _ { 3 } \sqrt { x - 1 }$. Since $81 = 3^4$, $\log _ { 3 } 81 = 4$. Also, $\log _ { 3 } \sqrt { x - 1 } = \frac { 1 } { 2 } \log _ { 3 } ( x - 1 )$. Thus, the expression simplifies to $4 - \frac { 1 } { 2 } \log _ { 3 } ( x - 1 )$.

Use Properties of Logarithms to Expand the Logarithmic Expression as Much