What is the answer to x=2^{x/2} ?

The answer to x=2^{x/2} is x=4,x=2

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

x=2^{x/2}

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

x = 4, x = 2

Solution steps

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

Prepare

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

for Lambert form:

x e^{- \frac{l n ( 2 ) x}{2}} = 1

Rewrite the equation with

- \frac{ln ( 2 ) x}{2} = u

and

x = - \frac{2 u}{ln ( 2 )}

(- \frac{2 u}{ln ( 2 )}) e^{u} = 1

Rewrite

(- \frac{2 u}{ln ( 2 )}) e^{u} = 1

in Lambert form:

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

Solve

e^{u} u = - \frac{ln ( 2 )}{2} : u = - 2 ln (2), u = - ln (2)

u = - 2 ln (2), u = - ln (2)

Substitute back

u = - \frac{ln ( 2 ) x}{2},

solve for

x

Solve

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

Solve

- \frac{ln ( 2 ) x}{2} = - ln (2) : x = 2

x = 4, x = 2

3^{5 x - 1} = 5^{2 x}

8^{x} = 132

4^{8 x - 5} = 1 6^{5 x + 7}

2^{x - 1} = 102 4^{x}

4^{x + 2} = 8^{x}