What is the answer to e^x-x^3=0 ?

The answer to e^x-x^3=0 is x=-3\W_{-1}(-1/3),x=-3\W_{0}(-1/3)

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e^x-x^3=0

e^{x} - x^{3} = 0

x = - 3 W_{- 1} (- \frac{1}{3}), x = - 3 W_{0} (- \frac{1}{3})

Solution steps

e^{x} - x^{3} = 0

Prepare

e^{x} - x^{3} = 0

for Lambert form:

1 = x e^{- \frac{x}{3}}

Rewrite the equation with

- \frac{x}{3} = u

and

x = - 3 u

1 = (- 3 u) e^{u}

Rewrite

1 = (- 3 u) e^{u}

in Lambert form:

e^{u} u = - \frac{1}{3}

Solve

e^{u} u = - \frac{1}{3} : u = W_{- 1} (- \frac{1}{3}), u = W_{0} (- \frac{1}{3})

u = W_{- 1} (- \frac{1}{3}), u = W_{0} (- \frac{1}{3})

Substitute back

u = - \frac{x}{3},

solve for

x

Solve

- \frac{x}{3} = W_{- 1} (- \frac{1}{3}) : x = - 3 W_{- 1} (- \frac{1}{3})

Solve

- \frac{x}{3} = W_{0} (- \frac{1}{3}) : x = - 3 W_{0} (- \frac{1}{3})

x = - 3 W_{- 1} (- \frac{1}{3}), x = - 3 W_{0} (- \frac{1}{3})

4^{3 x} = 8^{x - 1}

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

e^{5 x} = 4

3 x^{4} e^{- 2 l n (x)} + 2 x - 1 = 0

x^{x} = 2

What is the answer to e^x-x^3=0 ?
The answer to e^x-x^3=0 is x=-3\W_{-1}(-1/3),x=-3\W_{0}(-1/3)