What is the answer to x=e^{1/2 x-2} ?

The answer to x=e^{1/2 x-2} is x=-2\W_{-1}(-1/(2e^2)),x=-2\W_{0}(-1/(2e^2))

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x=e^{1/2 x-2}

x = e^{\frac{1}{2} x - 2}

x = - 2 W_{- 1} (- \frac{1}{2 e ^{2}}), x = - 2 W_{0} (- \frac{1}{2 e ^{2}})

Decimal Notation

x = 8.21093 \dots, x = 0.14555 \dots

Solution steps

x = e^{\frac{1}{2} x - 2}

Prepare

x = e^{\frac{1}{2} x - 2}

for Lambert form:

x e^{- \frac{1}{2} x + 2} = 1

Rewrite the equation with

- \frac{1}{2} x = u

and

x = - 2 u

(- 2 u) e^{- \frac{1}{2} (- 2 u) + 2} = 1

Simplify

- 2 e^{u + 2} u = 1

Rewrite

- 2 e^{u + 2} u = 1

in Lambert form:

e^{u} u = - \frac{1}{2 e ^{2}}

Solve

e^{u} u = - \frac{1}{2 e ^{2}} : u = W_{- 1} (- \frac{1}{2 e ^{2}}), u = W_{0} (- \frac{1}{2 e ^{2}})

u = W_{- 1} (- \frac{1}{2 e ^{2}}), u = W_{0} (- \frac{1}{2 e ^{2}})

Substitute back

u = - \frac{1}{2} x,

solve for

x

Solve

- \frac{1}{2} x = W_{- 1} (- \frac{1}{2 e ^{2}}) : x = - 2 W_{- 1} (- \frac{1}{2 e ^{2}})

Solve

- \frac{1}{2} x = W_{0} (- \frac{1}{2 e ^{2}}) : x = - 2 W_{0} (- \frac{1}{2 e ^{2}})

x = - 2 W_{- 1} (- \frac{1}{2 e ^{2}}), x = - 2 W_{0} (- \frac{1}{2 e ^{2}})

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

x, 35 - 12 e^{5 - 4 x} = 0

7^{w - 2} = 49

0. 1^{x} = 0.01

(1.05)^{x} = 2

What is the answer to x=e^{1/2 x-2} ?
The answer to x=e^{1/2 x-2} is x=-2\W_{-1}(-1/(2e^2)),x=-2\W_{0}(-1/(2e^2))