What is the answer to x^x=2^{64} ?

The answer to x^x=2^{64} is x=16

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x^x=2^{64}

x^{x} = 2^{64}

x = 16

Solution steps

x^{x} = 2^{64}

Prepare

x^{x} = 2^{64}

for Lambert form:

x e^{- \frac{64 l n ( 2 )}{x}} = 1

Rewrite the equation with

\frac{64 ln ( 2 )}{x} = u

and

x = \frac{64 ln ( 2 )}{u}

(\frac{64 ln ( 2 )}{u}) e^{- u} = 1

Rewrite

(\frac{64 ln ( 2 )}{u}) e^{- u} = 1

in Lambert form:

u e^{u} = 64 ln (2)

Solve

u e^{u} = 64 ln (2) : u = 4 ln (2)

Verify Solutions:

u = 4 ln (2)

True

The solution is

u = 4 ln (2)

Substitute back

u = \frac{64 ln ( 2 )}{x},

solve for

x

Solve

\frac{64 ln ( 2 )}{x} = 4 ln (2) : x = 16

x = 16

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