find-x-x-8-x-

Question Number 12446 by tawa last updated on 22/Apr/17

find x

\sqrt{x} = 8^{x}

Commented by mrW1 last updated on 23/Apr/17

F o r e q u a t i o n \sqrt{x} = a^{x} t h e s o l u t i o n i s

x = - \frac{W (- 2 \ln a)}{2 \ln a}

w h e r e 2 \ln a ⩽ \frac{1}{e} o r

a ⩽ e^{\frac{1}{2 e}} \approx 1.202

f o r a = 8 > 1.202 \Rightarrow t h e r e i s n o s o l u t i o n .

Answered by mrW1 last updated on 23/Apr/17

x ⩾ 0

x \overset{!}{=} 8^{2 x} = 64^{x}

f (x) = x

g (x) = 64^{x}

f (0) = 0

g (0) = 1 > f (0)

f^{'} (x) = 1

g^{'} (x) = 64^{x} \times \ln 64 > 1 = f^{'} (x)

\Rightarrow f o r a l l x ⩾ 0 w e h a v e g (x) > f (x)

i . e .

t h e r e i s n o s o l u t i o n f o r f (x) = g (x)!

Commented by mrW1 last updated on 22/Apr/17

Commented by tawa last updated on 23/Apr/17

God bless you sir

Commented by geovane10math last updated on 23/Apr/17

The solution \notin R

\in C

Answered by geovane10math last updated on 22/Apr/17

x = 8^{2 x}

x = 64^{x}

x = e^{x \cdot \ln 64}

\frac{x}{e^{x \cdot \ln 64}} = 1

x \cdot e^{- x \cdot \ln 64} = 1

(- \ln 64) (x \cdot e^{- \ln 64 \cdot x}) = - \ln 64

- \ln 64 \cdot x \cdot e^{- \ln 64 \cdot x} = - \ln 64

- \ln 64 \cdot x = y

{y e}^{y} = - \ln 64

y = W (- \ln 64) W : L a m b e r t f u n c t i o n

x = - \frac{W (- \ln 64)}{\ln 64}

Commented by mrW1 last updated on 22/Apr/17

W (- \ln 64) i s n o t v a l i d!

F o r L a m b e r t f u n c t i o n W (t)

t m u s t b e ⩾ - \frac{1}{e} \approx - 0.368

b u t - \ln 64 \approx - 4.159!!!

Commented by tawa last updated on 23/Apr/17

God bless you sir .

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