Rewrite the ratio as a ratio of natural logarithms and simplify.
-$\frac { \log _ { 243 } x } { \log 3 x }$
A) $\frac { 1 } { 5 }$
B) $3 ^ { 4 }$
C) $\ln 5$
D) $\frac { 1 } { 3 ^ { 4 } }$

Question

Quizplus · Accepted Answer

Using the change of base formula for logarithms, $\frac{\log_a b}{\log_c d} = \frac{\ln b}{\ln d}$, the given expression can be rewritten as $\frac{\ln x}{\ln 3x}$ for base 243 and 3 respectively. However, the correct simplification involves recognizing that 243 is $3^5$, leading to $\frac{\ln x}{5\ln 3 + \ln x}$. The question seems to have been misinterpreted in the explanation process. The correct simplification directly from the given expression, considering the properties of logarithms, should lead to a ratio that simplifies based on the relationship between the numbers 243 (which is $3^5$) and 3, not involving $\ln x$ in the numerator and denominator in the simplified form. The correct answer, considering the properties of logarithms and the change of base formula correctly, should reflect the relationship between the base numbers 243 and 3, leading to a simplified ratio that does not depend on x.

Rewrite the Ratio as a Ratio of Natural Logarithms and Simplify