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Solve

\(\large \color{red}{A}~=~ \sqrt{\dfrac{\log _{ 2015 }{ 2016 }}{\log _{ 2016 }{ 2015 }}}\)

 

\(\large {2015}^{\color{red}{A}} = \, ?\)

 Jul 8, 2020
 #1
avatar+25597 
+3

Solve
\(\large \color{red}{A}~=~ \sqrt{\dfrac{\log _{ 2015 }{ 2016 }}{\log _{ 2016 }{ 2015 }}}\)

 

\(\large {2015}^{\color{red}{A}} = \, ?\)

 

\(\begin{array}{|rcll|} \hline A &=& \sqrt{\dfrac{\log _{ 2015 }{ 2016 }} {\log _{ 2016 }{ 2015 }}} \\\\ && \boxed{ \log _{ 2015 }{ 2016 } = \dfrac{\ln(2016) } {\ln(2015)} \\~ \\ \log _{ 2016 }{ 2015 } = \dfrac{\ln(2015) } {\ln(2016)} } \\\\ A &=& \sqrt{ \dfrac{\ln(2016) } {\ln(2015)}\above 1pt \dfrac{\ln(2015) } {\ln(2016)} } \\\\ A &=& \sqrt{ \dfrac{\ln(2016)\ln(2016) } {\ln(2015)\ln(2015)} } \\\\ A &=& \sqrt{ \dfrac{\Big(\ln(2016)\Big)^2} {\Big(\ln(2015)\Big)^2} } \\\\ A &=& \dfrac{ \ln(2016) } { \ln(2015) } \\\\ A\ln(2015) &=& \ln(2016) \\ \ln(2015^A) &=& \ln(2016) \\ \mathbf{ 2015^A} &=& \mathbf{2016} \\ \hline \end{array}\)

 

 

laugh

 Jul 8, 2020

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