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Solve \(\log_x 3 = \log_{81} x\)

 Jun 17, 2020
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Solve 

\(\log_x (3) = \log_{81} (x)\)

 

\(\begin{array}{|rcll|} \hline \log_{81} (x) &=& \log_x (3) \\\\ x^{\log_{81} (x)} &=& x^{\log_x (3)} \quad | \quad x^{\log_x (3)}=3 \\\\ x^{\log_{81} (x)} &=& 3 \\ && \begin{array}{|rcll|} \hline \mathbf{x^{\log_{81} (x)}} &=& \mathbf{81^y} \quad | \quad log_{81} \text{ both sides } \\\\ log_{81}(x^{\log_{81} (x)}) &=& log_{81}(81^y) \quad | \quad log_{81}(81^y)=y \\\\ log_{81}(x^{\log_{81} (x)}) &=& y \\\\ \log_{81} (x)log_{81}(x) &=& y \\\\ \Big(\log_{81} (x)\Big)^2 &=& y \\\\ \mathbf{x^{\log_{81} (x)}} &=& \mathbf{ 81^{\Big(\log_{81} (x)\Big)^2} } \\ \hline \end{array} \\ 81^{\Big(\log_{81} (x)\Big)^2} &=& 3 \\ 3^{\color{red}4\Big(\log_{81} (x)\Big)^2} &=& 3^{\color{red}1} \\\\ \mathbf{4\Big(\log_{81} (x)\Big)^2} &=& \mathbf{1} \\\\ \Big(\log_{81} (x)\Big)^2 &=& \frac{1}{4} \\\\ \log_{81} (x) &=& \pm \sqrt{\frac{1}{4} } \\\\ \log_{81} (x) &=& \pm \frac{1}{2} \\\\ 81^{ \log_{81} (x)} &=& 81^{\pm \frac{1}{2}} \quad | \quad 81^{ \log_{81} (x)} =x \\\\ \mathbf{x} &=& \mathbf{81^{\pm \frac{1}{2}}} \\\\ x= 81^{\frac{1}{2}} &\text{or}& x= 81^{-\frac{1}{2}} \\\\ x= \sqrt{81} & & x= \dfrac{1}{\sqrt{81}} \\\\ \mathbf{x=9} & & \mathbf{ x= \dfrac{1}{9}} \\ \hline \end{array}\)

 

laugh

 Jun 17, 2020

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