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.
-$2 \log _ { 3 } 5 + \frac { 1 } { 3 } \log _ { 3 } ( r - 2 ) - \frac { 1 } { 2 } \log _ { 3 } r$
A) $\log _ { 3 } \frac { 25 \sqrt [ 3 ] { r - 2 } } { \sqrt { r } }$
B) $\log _ { 3 } \frac { 25 r - 2 } { 6 r }$
C) $\log _ { 3 } \sqrt { \frac { 2 r - 4 } { 6 r } }$
D) $\log _ { 3 } \left( \frac { 1 } { 3 } \sqrt { \frac { r - 2 } { r } } \right)$

Question

Quizplus · Accepted Answer

Using the properties of logarithms, the expression can be condensed as follows: The coefficient in front of a logarithm can be taken as an exponent inside the logarithm, so $2 \log _ { 3 } 5$ becomes $\log _ { 3 } 5^2$ or $\log _ { 3 } 25$. Similarly, $\frac { 1 } { 3 } \log _ { 3 } ( r - 2 )$ becomes $\log _ { 3 } (r - 2)^{\frac{1}{3}}$ or $\log _ { 3 } \sqrt[3]{r - 2}$, and $-\frac { 1 } { 2 } \log _ { 3 } r$ becomes $-\log _ { 3 } \sqrt{r}$. Adding and subtracting these logarithms according to the properties of logarithms (logarithm of a product for addition and logarithm of a quotient for subtraction) gives $\log _ { 3 } \frac { 25 \sqrt [ 3 ] { r - 2 } } { \sqrt { r } }$.

Use Properties of Logarithms to Condense the Logarithmic Expression 2log⁡35+13log⁡3(r−2)−12log⁡3r2 \log _ { 3 } 5 + \frac { 1 } { 3 } \log _ { 3 } ( r - 2 ) - \frac { 1 } { 2 } \log _ { 3 } r2log3​5+31​log3​(r−2)−21​log3​r

Use Properties of Logarithms to Condense the Logarithmic Expression $2 \log _ { 3 } 5 + \frac { 1 } { 3 } \log _ { 3 } ( r - 2 ) - \frac { 1 } { 2 } \log _ { 3 } r$