en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

x^e=e^x

Solution

x^{e} = e^{x}

Solution

x = e

+1

Decimal Notation

x = 2.71828 \dots

Solution steps

x^{e} = e^{x}

Prepare

x^{e} = e^{x}

for Lambert form:

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

Rewrite the equation with

- \frac{x}{e} = u

and

x = - e u

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

Rewrite

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

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

roots ae^x roots

a e^{x}

4^{3 x + 1} = 16

3^{2(x+1)}-18\times 3^x+9=0

3^{2 (x + 1)} - 18 \times 3^{x} + 9 = 0

6^{3m}*6^{-m}=6^{-2m}

6^{3 m} \cdot 6^{- m} = 6^{- 2 m}

100 = 50 e^{10 x}

Frequently Asked Questions (FAQ)

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