Choose the one alternative that best completes the statement or answers the question.
-$\log \left( 1 - \frac { 1 } { x ^ { 3 } } \right)$
A) $\log ( x - 1 ) + \log \left( x ^ { 2 } + 1 \right) - 3 \log x$
B) $\log 1 - 3 \log 1 - 3 \log x$
C) $\log ( x - 1 ) + \log \left( x ^ { 2 } + x + 1 \right) - 3 \log x$
D) $\log x ^ { 3 } - \log 1 - 3 \log x$

Question

Quizplus · Accepted Answer

The expression $\log \left( 1 - \frac { 1 } { x ^ { 3 } } \right)$ can be rewritten using properties of logarithms and the fact that $1 - \frac{1}{x^3}$ can be factored as $(x^3 - 1)/x^3$, which further factors into $(x-1)(x^2+x+1)/x^3$. Applying the logarithm properties, this becomes $\log(x-1) + \log(x^2+x+1) - 3\log(x)$, matching choice C.

Choose the One Alternative That Best Completes the Statement or Answers