Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.
-$\log _ { 5 } \frac { 125 } { \sqrt { x - 1 } }$
A) $3 - \log _ { 5 } \sqrt { x - 1 }$
B) $3 \log _ { 5 } 5 - \frac { 1 } { 2 } \log _ { 5 } ( x - 1 )$
C) $\log _ { 5 } 125 - \log _ { 5 } \sqrt { x - 1 }$
D) $3 - \frac { 1 } { 2 } \log _ { 5 } ( x - 1 )$

Question

Quizplus · Accepted Answer

The expression can be expanded using the properties of logarithms: $\log _ { 5 } \frac { 125 } { \sqrt { x - 1 } } = \log _ { 5 } 125 - \log _ { 5 } \sqrt { x - 1 }$. Since $\log _ { 5 } 125 = 3$ (because $5^3 = 125$) and $\log _ { 5 } \sqrt { x - 1 } = \frac { 1 } { 2 } \log _ { 5 } ( x - 1 )$ (using the property $\log_b \sqrt[n]{x} = \frac{1}{n} \log_b x$), the expression simplifies to $3 - \frac { 1 } { 2 } \log _ { 5 } ( x - 1 )$.

Use Properties of Logarithms to Expand the Logarithmic Expression as Much