Choose the one alternative that best completes the statement or answers the question.
-$\ln \frac { ( 6 x ) \sqrt [ 9 ] { 1 + 3 x } } { ( x - 9 ) ^ { 7 } } , \quad x 9$
A) $6 \ln x + \frac { 1 } { 3 } \ln ( 1 + 3 x ) - 7 \ln ( x - 9 )$
B) $\ln 6 + \ln x + \frac { 1 } { 9 } \ln ( 1 + 3 x ) - 7 \ln ( x - 9 )$
C) $\ln 6 + \ln x - 9 \ln ( 1 + 3 x ) - 7 \ln ( x - 9 )$
D) $\ln 6 + \ln x + \frac { 1 } { 9 } \ln ( 1 + 3 x ) - \ln 7 - \ln ( x - 9 )$

Question

Choose the one alternative that best completes the statement or answers the question.
-$\ln \frac { ( 6 x ) \sqrt [ 9 ] { 1 + 3 x } } { ( x - 9 ) ^ { 7 } } , \quad x > 9$
A) $6 \ln x + \frac { 1 } { 3 } \ln ( 1 + 3 x ) - 7 \ln ( x - 9 )$
B) $\ln 6 + \ln x + \frac { 1 } { 9 } \ln ( 1 + 3 x ) - 7 \ln ( x - 9 )$
C) $\ln 6 + \ln x - 9 \ln ( 1 + 3 x ) - 7 \ln ( x - 9 )$
D) $\ln 6 + \ln x + \frac { 1 } { 9 } \ln ( 1 + 3 x ) - \ln 7 - \ln ( x - 9 )$

Quizplus · Accepted Answer

Using the properties of logarithms, the expression can be simplified as follows: $\ln \frac { ( 6 x ) \sqrt [ 9 ] { 1 + 3 x } } { ( x - 9 ) ^ { 7 } }$ can be broken down into $\ln 6 + \ln x + \frac { 1 } { 9 } \ln ( 1 + 3 x ) - 7 \ln ( x - 9 )$, which matches choice B. This uses the properties that $\ln ab = \ln a + \ln b$, $\ln a^b = b \ln a$, and $\ln \frac{a}{b} = \ln a - \ln b$.

Choose the One Alternative That Best Completes the Statement or Answers