en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

e^{-x}-x+1=0

Solution

e^{- x} - x + 1 = 0

Solution

x = W_{0} (\frac{1}{e}) + 1

+1

Decimal Notation

x = 1.27846 \dots

Solution steps

e^{- x} - x + 1 = 0

Prepare

e^{- x} - x + 1 = 0

for Lambert form:

1 = - e^{x} (- x + 1)

Rewrite the equation with

x - 1 = u

and

x = u + 1

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

Simplify

1 = e^{u + 1} u

Rewrite

1 = e^{u + 1} u

in Lambert form:

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

Solve

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

u = W_{0} (\frac{1}{e})

Substitute back

u = x - 1,

solve for

x

Solve

x - 1 = W_{0} (\frac{1}{e}) : x = W_{0} (\frac{1}{e}) + 1

x = W_{0} (\frac{1}{e}) + 1

Popular Examples

9^{x+1}-28*3^x+3=0

9^{x + 1} - 28 \cdot 3^{x} + 3 = 0

3 \cdot 6^{1 - x} = 15

500e^{3x+2}=15000

500 e^{3 x + 2} = 15000

(1/9)^{x-5}=3^{3x}

(\frac{1}{9})^{x - 5} = 3^{3 x}

4^{x-x^2}= 1/(16^x)

4^{x - x^{2}} = \frac{1}{1 6 ^{x}}

Frequently Asked Questions (FAQ)

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