Question

prove the equation below.

Answer 1

\[\frac{4^\frac{1}{\log_{4}\frac{3}{4}}}{3^\frac{1}{\log_{3}\frac{3}{4}}} = \frac{1}{12}\]

Answer 2

\[\frac{1}{\log_{4}{ \frac{3}4}}=\log_{\frac{3}4}4=\log_{\frac{3}4}{(3*\frac{4}3)}=\log_{\frac{3}4}{3}+\log_{\frac{3}4}{\frac{4}3}\]
and \[\frac{1}{\log_{3}{\frac{3}4}}=\log_{\frac{3}4}3\]
so the equation equals to:\[\frac{ 4^{\log_{ \frac{3}4 }4 } } {3^{\log_{\frac{3}4}3}}=\frac{ 4^{ \log_{\frac{3}4}3 + \log_{\frac{3}4}{\frac{4}3}} }{ 3^{\log_{\frac{3}4}3} }=\frac{ 4^{ \log_{\frac{3}4}3} * 4^{ \log_{\frac{3}4}{\frac{4}3}} }{ 3^{\log_{\frac{3}4}3} } = (\frac{4}3)^{ \log_{\frac{3}4}3} *4^{\log_{\frac{3}4}{\frac{4}3}}=\frac{1}{ {\frac{3}4}^{\log_{\frac{3}4}3}}*4^{-1}=\frac{1}3*\frac{1}4\]
\[=\frac{1}{12}\]