What is the answer to x^{sqrt(x)}=2 ?

The answer to x^{sqrt(x)}=2 is x=(ln^2(2)}{4\W_{0}(\frac{ln(2))/2)^2}

Solve for x: x^{sqrt(x)}=2

The solution is x=(ln^2(2)}{4\W_{0}(\frac{ln(2))/2)^2}

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x^{x} = 2

x = \frac{ln ^{2} ( 2 )}{4 W _{0} ( \frac{l n ( 2 )}{2} ) ^{2}}

Solution steps

x^{x} = 2

Prepare

x^{x} = 2

for Lambert form:

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

Rewrite the equation with

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

and

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

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

Solve

(\frac{ln ^{2} ( 2 )}{u ^{2}}) e^{- u} = 1 : u = - 4 ln (2), u = - 2 ln (2), u = 2 W_{0} (\frac{ln ( 2 )}{2})

u = - 4 ln (2), u = - 2 ln (2), u = 2 W_{0} (\frac{ln ( 2 )}{2})

Substitute back

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

solve for

x

Solve

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

No Solution for

x \in R

Solve

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

No Solution for

x \in R

Solve

\frac{ln ( 2 )}{x} = 2 W_{0} (\frac{ln ( 2 )}{2}) : x = \frac{ln ^{2} ( 2 )}{4 W _{0} ( \frac{l n ( 2 )}{2} ) ^{2}}

x = \frac{ln ^{2} ( 2 )}{4 W _{0} ( \frac{l n ( 2 )}{2} ) ^{2}}

Verify Solutions:

x = \frac{ln ^{2} ( 2 )}{4 W _{0} ( \frac{l n ( 2 )}{2} ) ^{2}}

True

The solution is

x = \frac{ln ^{2} ( 2 )}{4 W _{0} ( \frac{l n ( 2 )}{2} ) ^{2}}

Verify Solutions:

x = \frac{ln ^{2} ( 2 )}{4 W _{0} ( \frac{l n ( 2 )}{2} ) ^{2}}

True

The solution is

x = \frac{ln ^{2} ( 2 )}{4 W _{0} ( \frac{l n ( 2 )}{2} ) ^{2}}

4^{3 x - 1} = (0.5)^{- x - 3}

e^{- x^{2} - y^{2}} x (1 - 2 y^{2}) = 0

\frac{n}{2 ^{x}} = 1

550 \cdot 3^{x} = 50

\frac{8 ^{x} + 1}{8 ^{x}} = 9

What is the answer to x^{sqrt(x)}=2 ?
The answer to x^{sqrt(x)}=2 is x=(ln^2(2)}{4\W_{0}(\frac{ln(2))/2)^2}
Solve for x: x^{sqrt(x)}=2
The solution is x=(ln^2(2)}{4\W_{0}(\frac{ln(2))/2)^2}