Combine into a single logarithm. $\log ( x - 3 ) + \log ( x + 3 ) - \frac { 1 } { 2 } \log \left( x ^ { 2 } + 9 \right)$

A) $\log \left( \frac { x ^ { 2 } - 9 } { x ^ { 2 } + 9 } \right)$

B) $\log \left( \frac { \sqrt { x ^ { 2 } - 9 } } { x ^ { 2 } } + 9 \right)$

C) $\log \left( \frac { x ^ { 2 } - 9 } { \sqrt { x ^ { 2 } + 9 } } \right)$

D) $\log \left( \frac { x - 3 } { x + 3 } \right)$

E) $\log \left( \frac { x - 9 } { \sqrt { x + 9 } } \right)$

Question

Combine into a single logarithm. $\log ( x - 3 ) + \log ( x + 3 ) - \frac { 1 } { 2 } \log \left( x ^ { 2 } + 9 \right)$

A) $\log \left( \frac { x ^ { 2 } - 9 } { x ^ { 2 } + 9 } \right)$

B) $\log \left( \frac { \sqrt { x ^ { 2 } - 9 } } { x ^ { 2 } } + 9 \right)$

C) $\log \left( \frac { x ^ { 2 } - 9 } { \sqrt { x ^ { 2 } + 9 } } \right)$

D) $\log \left( \frac { x - 3 } { x + 3 } \right)$

E) $\log \left( \frac { x - 9 } { \sqrt { x + 9 } } \right)$

Quizplus · Accepted Answer

Using the logarithmic rule $\log a+\log b = \log(ab)$, we can rewrite the expression as $\log((x-3)(x+3))- \frac{1}{2}\log(x^2+9)$. Then, using the logarithmic rule $\log a^n = n\log a$, we can rewrite the expression again as $\log\left(\frac{(x-3)(x+3)}{\sqrt{x^2+9}}\right)$. Simplifying the expression inside the logarithm using the difference of squares, we get $\log\left(\frac{x^2-9}{\sqrt{x^2+9}}\right)$. Factoring out the $\sqrt{x^2+9}$ in the denominator, we get $\log\left(\frac{x^2-9}{\sqrt{x^2+9}}\right)$, which matches answer choice C. Therefore, the best choice is C.

Combine into a Single Logarithm log⁡(x−3)+log⁡(x+3)−12log⁡(x2+9)\log ( x - 3 ) + \log ( x + 3 ) - \frac { 1 } { 2 } \log \left( x ^ { 2 } + 9 \right)log(x−3)+log(x+3)−21​log(x2+9)

Combine into a Single Logarithm $\log ( x - 3 ) + \log ( x + 3 ) - \frac { 1 } { 2 } \log \left( x ^ { 2 } + 9 \right)$