find-x-2-x-8x-

Question Number 187856 by Michaelfaraday last updated on 23/Feb/23

f i n d x

2^{\sqrt{x}} = 8 x

Answered by mr W last updated on 23/Feb/23

t = \sqrt{x} > 0

2^{t} = 8 t^{2}

2^{\frac{t}{2}} = 2 \sqrt{2} t

e^{\frac{t \ln 2}{2}} = 4 \sqrt{2} \times \frac{t}{2}

- \frac{t \ln 2}{2} e^{- \frac{t \ln 2}{2}} = - \frac{\ln 2}{4 \sqrt{2}}

- \frac{t \ln 2}{2} = W (- \frac{\ln 2}{4 \sqrt{2}})

t = - \frac{2}{\ln 2} W (- \frac{\ln 2}{4 \sqrt{2}})

x = t^{2} = {[- \frac{2}{\ln 2} W (- \frac{\ln 2}{4 \sqrt{2}})]}^{2}

\approx {\begin{cases} {[- \frac{2 \times (- 3.29038565)}{\ln 2}]}^{2} = 90.136912 \\ {[- \frac{2 \times (- 0.141101)}{\ln 2}]}^{2} = 0.165756 \end{cases}

Commented by Michaelfaraday last updated on 25/Feb/23

t h a n k s s i r