4-1-x-6-1-x-9-1-x-x-R-

Question Number 187672 by Humble last updated on 20/Feb/23

4^{- \frac{1}{x}} + 6^{- \frac{1}{x}} = 9^{- \frac{1}{x}}

x \in R

Answered by SEKRET last updated on 20/Feb/23

\frac{4^{- \frac{1}{\boldsymbol x}}}{9^{- \frac{1}{\boldsymbol x}}} + \frac{6^{\frac{- 1}{\boldsymbol x}}}{9^{\frac{- 1}{\boldsymbol x}}} = 1

{(\frac{4}{9})}^{\frac{- 1}{\boldsymbol x}} + {(\frac{6}{9})}^{\frac{- 1}{\boldsymbol x}} = 1

{(\frac{2}{3})}^{2 \cdot \frac{- 1}{\boldsymbol x}} + {(\frac{2}{3})}^{\frac{- 1}{\boldsymbol x}} = 1

\boldsymbol t^{2} + \boldsymbol t = 1

\boldsymbol t^{2} + 2 \cdot \frac{1}{2} \cdot \boldsymbol t + \frac{1}{4} = 1 + \frac{1}{4}

{(\boldsymbol t + \frac{1}{2})}^{2} = \frac{5}{4}

\boldsymbol t = \frac{- 1 \mp \sqrt{5}}{2}

{(\frac{2}{3})}^{\frac{- 1}{\boldsymbol x}} = \frac{\sqrt{5} - 1}{2}

\frac{- 1}{\boldsymbol x} = \boldsymbol \log_{(\frac{2}{3})} (\frac{\sqrt{5} - 1}{2})

\frac{1}{\boldsymbol x} = \boldsymbol \log_{(\frac{3}{2})} (\frac{\sqrt{5} - 1}{2})

\boldsymbol x = \boldsymbol \log_{(\frac{\sqrt{5} - 1}{2})} (\frac{3}{2})

Commented by Humble last updated on 20/Feb/23

n i c e s o l u t i o n, s i r

Answered by Rasheed.Sindhi last updated on 20/Feb/23

4^{- \frac{1}{x}} + 6^{- \frac{1}{x}} = 9^{- \frac{1}{x}}

{(\frac{4}{6})}^{- 1 / x} + 1 = {(\frac{9}{6})}^{- 1 / x}

{(\frac{3}{2})}^{1 / x} + 1 = {(\frac{2}{3})}^{1 / x}

y + 1 = \frac{1}{y}; y ⩾ 0

y^{2} + y - 1 = 0

y = \frac{- 1 + \sqrt{1 + 4}}{2}

\log {(\frac{3}{2})}^{1 / x} = \log (\frac{- 1 + \sqrt{5}}{2})

(\frac{1}{x}) \log (\frac{3}{2}) = \log (- 1 + \sqrt{5}) - \log 2

x = \frac{\log 3 - \log 2}{\log (- 1 + \sqrt{5}) - \log 2}

Commented by Humble last updated on 20/Feb/23

E x c e l l e n t!!

E x c e l l e n t!!

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