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Popular >

x^a=e^{x-1}

Solution

x^{a} = e^{x - 1}

Solution

x = - a W (- \frac{1}{a e ^{\frac{1}{a}}})

Solution steps

x^{a} = e^{x - 1}

Prepare

x^{a} = e^{x - 1}

for Lambert form:

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

Rewrite the equation with

- \frac{x}{a} = u

and

x = - a u

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

Simplify

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

Rewrite

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

in Lambert form:

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

Solve

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

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

Substitute back

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

solve for

x

Solve

- \frac{x}{a} = W (- \frac{1}{a e ^{\frac{1}{a}}}) : x = - a W (- \frac{1}{a e ^{\frac{1}{a}}})

x = - a W (- \frac{1}{a e ^{\frac{1}{a}}})

Popular Examples

solvefor y,(x^{-2})/(x^y)= 1/(x^{12)}solve for

y, \frac{x ^{- 2}}{x ^{y}} = \frac{1}{x ^{12}}

2^{2 x - 1} = 4^{x} - 2

9^{x} - 6 = - 3^{x}

d^{2} - 4 d + 4 = 2^{x}

(60.431)^a=10^e

(60.431)^{a} = 1 0^{e}

Frequently Asked Questions (FAQ)

What is the answer to x^a=e^{x-1} ?
The answer to x^a=e^{x-1} is x=-a\W(-1/(ae^{1/a)})