What is the answer to 10xe^{1-x}=1 ?

The answer to 10xe^{1-x}=1 is x=-\W_{-1}(-1/(10e)),x=-\W_{0}(-1/(10e))

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10xe^{1-x}=1

10 x e^{1 - x} = 1

x = - W_{- 1} (- \frac{1}{10 e}), x = - W_{0} (- \frac{1}{10 e})

Decimal Notation

x = 4.88972 \dots, x = 0.03822 \dots

Solution steps

10 x e^{1 - x} = 1

Rewrite the equation with

- x = u

and

x = - u

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

Simplify

- 10 e^{1 + u} u = 1

Rewrite

- 10 e^{1 + u} u = 1

in Lambert form:

e^{u} u = - \frac{1}{10 e}

Solve

e^{u} u = - \frac{1}{10 e} : u = W_{- 1} (- \frac{1}{10 e}), u = W_{0} (- \frac{1}{10 e})

u = W_{- 1} (- \frac{1}{10 e}), u = W_{0} (- \frac{1}{10 e})

Substitute back

u = - x,

solve for

x

Solve

- x = W_{- 1} (- \frac{1}{10 e}) : x = - W_{- 1} (- \frac{1}{10 e})

Solve

- x = W_{0} (- \frac{1}{10 e}) : x = - W_{0} (- \frac{1}{10 e})

x = - W_{- 1} (- \frac{1}{10 e}), x = - W_{0} (- \frac{1}{10 e})

9^{x + 1} = 2 7^{x - 1}

4^{x - 11} = \frac{1}{16}

4^{x} + 2^{x + 1} - 1 = 0

1 + e^{3 x - 1} = 20

4^{2 x - 1} = \frac{1}{64}

What is the answer to 10xe^{1-x}=1 ?
The answer to 10xe^{1-x}=1 is x=-\W_{-1}(-1/(10e)),x=-\W_{0}(-1/(10e))