Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions.
-$5 \log _ { 6 } 5 + \frac { 1 } { 4 } \log _ { 6 } ( r - 2 ) - \frac { 1 } { 2 } \log _ { 6 } r$
A) $\log _ { 6 } \frac { 3125 r - 2 } { 8 r }$
B) $\log _ { 6 } \frac { 3125 \sqrt [ 4 ] { \mathrm { r } - 2 } } { \sqrt { \mathrm { r } } }$
C) $\log _ { 6 } \sqrt { \frac { 5 r - 10 } { 8 r } }$
D) $\log _ { 6 } \left( \frac { 5 } { 8 } \sqrt { \frac { r - 2 } { r } } \right)$

Question

Quizplus · Accepted Answer

The expression can be condensed using the properties of logarithms: $5\log_6 5$ can be written as $\log_6 5^5$, $\frac{1}{4}\log_6(r-2)$ as $\log_6(r-2)^{\frac{1}{4}}$, and $-\frac{1}{2}\log_6 r$ as $\log_6 r^{-\frac{1}{2}}$. Combining these using the product and quotient rules for logarithms results in $\log_6 \frac{3125\sqrt[4]{r-2}}{\sqrt{r}}$.

Use Properties of Logarithms to Condense the Logarithmic Expression 5log⁡65+14log⁡6(r−2)−12log⁡6r5 \log _ { 6 } 5 + \frac { 1 } { 4 } \log _ { 6 } ( r - 2 ) - \frac { 1 } { 2 } \log _ { 6 } r5log6​5+41​log6​(r−2)−21​log6​r

Use Properties of Logarithms to Condense the Logarithmic Expression $5 \log _ { 6 } 5 + \frac { 1 } { 4 } \log _ { 6 } ( r - 2 ) - \frac { 1 } { 2 } \log _ { 6 } r$