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Popular >

e^x=x^e

Solution

e^{x} = x^{e}

Solution

x = e

+1

Decimal Notation

x = 2.71828 \dots

Solution steps

e^{x} = x^{e}

Prepare

e^{x} = x^{e}

for Lambert form:

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

Rewrite the equation with

- \frac{x}{e} = u

and

x = - e u

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

Rewrite

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

in Lambert form:

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

Solve

e^{u} u = - \frac{1}{e} : u = - 1

u = - 1

Substitute back

u = - \frac{x}{e},

solve for

x

Solve

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

x = e

Verify Solutions:

x = e

True

The solution is

x = e

Verify Solutions:

x = e

True

The solution is

x = e

Popular Examples

2^{x - 3} = 0.5

e^{(-0.005t)}=100

e^{(- 0.005 t)} = 100

(1/8)^{x+1}=32^x

(\frac{1}{8})^{x + 1} = 3 2^{x}

0. 5^{x} = 10

9^{y - 1} = 3

Frequently Asked Questions (FAQ)

What is the answer to e^x=x^e ?
The answer to e^x=x^e is x=e