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

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

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e^x=x^3

e^{x} = x^{3}

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

Decimal Notation

x = 4.53640 \dots, x = 1.85718 \dots

Solution steps

e^{x} = x^{3}

Prepare

e^{x} = x^{3}

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})

\frac{16 e ^{x}}{3 e ^{x} + 6} = 2

1 = 0. 5^{x}

2^{x - 1} = 8

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

2^{x} = 125

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