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Popular >

x^x=e

Solution

x^{x} = e

Solution

x = \frac{1}{W _{0} ( 1 )}

+1

Decimal Notation

x = 1.76322 \dots

Solution steps

x^{x} = e

Prepare

x^{x} = e

for Lambert form:

x e^{- \frac{1}{x}} = 1

Rewrite the equation with

\frac{1}{x} = u

and

x = \frac{1}{u}

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

Rewrite

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

in Lambert form:

u e^{u} = 1

Solve

u e^{u} = 1 : u = W_{0} (1)

Verify Solutions:

u = W_{0} (1)

True

The solution is

u = W_{0} (1)

Substitute back

u = \frac{1}{x},

solve for

x

Solve

\frac{1}{x} = W_{0} (1) : x = \frac{1}{W _{0} ( 1 )}

x = \frac{1}{W _{0} ( 1 )}

Popular Examples

51 e^{3 x} = 102

9^{x} = 3^{x + 5}

ln(e^{3+2ln(x)})=58

ln (e^{3 + 2 l n (x)}) = 58

3^{- x} = 27

(\frac{1}{27}) = 3^{x}

Frequently Asked Questions (FAQ)

What is the answer to x^x=e ?
The answer to x^x=e is x= 1/(\W_{0)(1)}