What is the answer to x^x=8 ?

The answer to x^x=8 is x=(3ln(2))/(\W_{0)(3ln(2))}

The solution is x=(3ln(2))/(\W_{0)(3ln(2))}

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x^x=8

x^{x} = 8

x = \frac{3 ln ( 2 )}{W _{0} ( 3 ln ( 2 ) )}

Solution steps

x^{x} = 8

Prepare

x^{x} = 8

for Lambert form:

x e^{- \frac{3 l n ( 2 )}{x}} = 1

Rewrite the equation with

\frac{3 ln ( 2 )}{x} = u

and

x = \frac{3 ln ( 2 )}{u}

(\frac{3 ln ( 2 )}{u}) e^{- u} = 1

Rewrite

(\frac{3 ln ( 2 )}{u}) e^{- u} = 1

in Lambert form:

u e^{u} = 3 ln (2)

Solve

u e^{u} = 3 ln (2) : u = W_{0} (3 ln (2))

Verify Solutions:

u = W_{0} (3 ln (2))

True

The solution is

u = W_{0} (3 ln (2))

Substitute back

u = \frac{3 ln ( 2 )}{x},

solve for

x

Solve

\frac{3 ln ( 2 )}{x} = W_{0} (3 ln (2)) : x = \frac{3 ln ( 2 )}{W _{0} ( 3 ln ( 2 ) )}

x = \frac{3 ln ( 2 )}{W _{0} ( 3 ln ( 2 ) )}

x^{x} = 7

2 x = e^{- x}

e^{- l n (x)} = 20

2^{6} \cdot 2^{12} = (2^{x})^{2}

2^{x} = \frac{32}{8}

What is the answer to x^x=8 ?
The answer to x^x=8 is x=(3ln(2))/(\W_{0)(3ln(2))}
Solve for x: x^x=8
The solution is x=(3ln(2))/(\W_{0)(3ln(2))}