find-x-if-4-x-6-x-9-x-findx-

Question Number 96242 by joki last updated on 30/May/20

find x if 4^{x} + 6^{x} = 9^{x} findx ?

Answered by john santu last updated on 30/May/20

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

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

t^{2} + t - 1 = 0, t = {(\frac{2}{3})}^{x}

t = \frac{- 1 + \sqrt{5}}{2} \Rightarrow x = \log_{(\frac{2}{3})} (\frac{\sqrt{5} - 1}{2})

Answered by 1549442205 last updated on 31/May/20

Dividing two sides of the equation by

6^{x} we get {(\frac{2}{3})}^{x} + 1 = {(\frac{3}{2})}^{x} . Putting {(\frac{2}{3})}^{x}

= t (t > 0), we obtain : t - \frac{1}{t} + 1 = 0 \Leftrightarrow t^{2} + t - 1 = 0

t = \frac{- 1 + \sqrt{5}}{2} . From that {(\frac{2}{3})}^{x} = \frac{\sqrt{5} - 1}{2}

xln (\frac{2}{3}) = \ln \frac{\sqrt{5} - 1}{2} \Rightarrow x = \frac{\ln \frac{\sqrt{5} - 1}{2}}{\ln \frac{2}{3}} \approx 1.1868