Condense the expression below to the logarithm of a single quantity.
$$5 [ \ln x - \ln ( x + 2 ) - \ln ( x - 2 ) ]$$
A) $\quad \ln \left( x ^ { 5 } - \left( x ^ { 2 } - 4 \right) ^ { 5 } \right)$

B) $\quad \ln \left( x ^ { 5 } - ( x + 2 ) ^ { 5 } - ( x - 2 ) ^ { 5 } \right)$

C) $\ln \frac { x ^ { 5 } } { x ^ { 2 } - 4 }$

D) $\ln \left( \left( \frac { x } { x ^ { 2 } - 4 } \right) ^ { 5 } \right)$

E) $\quad \ln \frac { x } { \left( x ^ { 2 } - 4 \right) ^ { 5 } }$

Question

Condense the expression below to the logarithm of a single quantity.
$$5 [ \ln x - \ln ( x + 2 ) - \ln ( x - 2 ) ]$$
A) $\quad \ln \left( x ^ { 5 } - \left( x ^ { 2 } - 4 \right) ^ { 5 } \right)$

B) $\quad \ln \left( x ^ { 5 } - ( x + 2 ) ^ { 5 } - ( x - 2 ) ^ { 5 } \right)$

C) $\ln \frac { x ^ { 5 } } { x ^ { 2 } - 4 }$

D) $\ln \left( \left( \frac { x } { x ^ { 2 } - 4 } \right) ^ { 5 } \right)$

E) $\quad \ln \frac { x } { \left( x ^ { 2 } - 4 \right) ^ { 5 } }$

Quizplus · Accepted Answer

Using the properties of logarithms, the given expression can be simplified as follows: $$5 [ \ln x - \ln ( x + 2 ) - \ln ( x - 2 ) ] = 5 \ln \left( \frac{x}{(x+2)(x-2)} \right) = \ln \left( \left( \frac{x}{x^2-4} \right)^5 \right).$$ This matches choice D.

Condense the Expression Below to the Logarithm of a Single 5[ln⁡x−ln⁡(x+2)−ln⁡(x−2)]5 [ \ln x - \ln ( x + 2 ) - \ln ( x - 2 ) ]5[lnx−ln(x+2)−ln(x−2)]

Condense the Expression Below to the Logarithm of a Single $5 [ \ln x - \ln ( x + 2 ) - \ln ( x - 2 ) ]$