solve-2-x-4x-

Question Number 172111 by Mikenice last updated on 23/Jun/22

s o l v e

2^{x} = 4 x

Commented by mr W last updated on 23/Jun/22

y o u h a v e a s k e d 2^{x} = 10 x . i t i s s o l v e d

g e n e r a l l y f o r a^{x} = b x . s o p l e a s e {d o n}^{'} t

p o s t s i m i l a r q u e s t i o n s l i k e

2^{x} = 4 x,

2^{x} = 5 x e t c \dots

Answered by Mikenice last updated on 23/Jun/22

l e t 2^{x} = p

\Rightarrow x = \sqrt{p}

p = 4 \sqrt{p}

t a k e s q u a r e t o b o t h s i d e

p^{2} = 16 p

p = 16, t h e n

2^{x} = 16

2^{x} = 2^{4}

x = 4 .

Commented by mr W last updated on 23/Jun/22

x = 0.3099 i s s o l u t i o n t o o .

Commented by mr W last updated on 23/Jun/22

p l e a s e c h e c k f o l l o w i n g s t e p s i r :

l e t 2^{x} = p

\Rightarrow x = \sqrt{p} s h o u l d i t n o t b e x = \log_{2} p ?

Commented by Mikenice last updated on 23/Jun/22

n o, t h a t i s w h a t i t w i l l b e b u t y o u c a n t r y i t b y u s i n g i n

y o u r o w n w a y

Commented by mr W last updated on 24/Jun/22

2^{x} = p \Rightarrow x = \sqrt{p} i s w r o n g .

n o t i s a y t h a t, b u t t h e m a t h e m a t i c s

s a y s t h a t .

I T I S N O T P E R S O N A L s i r .

Answered by Mikenice last updated on 24/Jun/22

l e t 2^{x} = y - - - 1

x = {l o g}_{2} y - - - 2

e q u a t e e q n 2 b y e q n 1

y = 4 {l o g}_{2} y, t h e n d i v i d e b o t h s i d e b y 4

\frac{y}{4} = {l o g}_{2} y \Rightarrow y = 2^{\frac{y}{4}} - - - 3

s u b e q n 3 i n t o e q n 1

2^{x} = 2^{\frac{y}{4}}

x = \frac{y}{4} - - - 4

l e t y = 16, t h e n

x = \frac{16}{4} \Rightarrow x = 4 .

Answered by mr W last updated on 25/Jun/22

2^{x} = 4 x

e^{x \ln 2} = 4 x

(- x \ln 2) e^{- x \ln 2} = - \frac{\ln 2}{4}

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

\Rightarrow x = - \frac{W (- \frac{\ln 2}{4})}{\ln 2}

= {\begin{cases} - \frac{- 2.77258872}{\ln 2} = 4 \\ - \frac{- 0.21481112}{\ln 2} = 0.309907 \end{cases}

g e n e r a l l y f o r

a^{x} = b x

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

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