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

Question

Quizplus · Accepted Answer

The given ratio can be rewritten using the change of base formula for logarithms: $\frac{\log_a x}{\log_b x} = \frac{\ln x / \ln a}{\ln x / \ln b} = \frac{\ln b}{\ln a}$. Applying this to the given ratio, we get $\frac{\ln(\sqrt{3})}{\ln(\sqrt{5})} = \frac{\ln 3}{\ln 5}$.

Rewrite the Ratio as a Ratio of Natural Logarithms and Simplify