What is the answer to x^x=3^{81} ?

The answer to x^x=3^{81} is x=27

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x^x=3^{81}

x^{x} = 3^{81}

x = 27

Solution steps

x^{x} = 3^{81}

Prepare

x^{x} = 3^{81}

for Lambert form:

x e^{- \frac{81 l n ( 3 )}{x}} = 1

Rewrite the equation with

\frac{81 ln ( 3 )}{x} = u

and

x = \frac{81 ln ( 3 )}{u}

(\frac{81 ln ( 3 )}{u}) e^{- u} = 1

Rewrite

(\frac{81 ln ( 3 )}{u}) e^{- u} = 1

in Lambert form:

u e^{u} = 81 ln (3)

Solve

u e^{u} = 81 ln (3) : u = 3 ln (3)

Verify Solutions:

u = 3 ln (3)

True

The solution is

u = 3 ln (3)

Substitute back

u = \frac{81 ln ( 3 )}{x},

solve for

x

Solve

\frac{81 ln ( 3 )}{x} = 3 ln (3) : x = 27

x = 27

- e^{x} - x e^{x} = 0

e^{x} = - 9 x

2 7^{x} - 3^{x + 2} = 9^{x} - 9

1 0^{x + 2} - 1 0^{x} = 9900

2 e^{2 x} + 2 e^{- 2 x} - e^{x} + e^{- x} = 7