What is the answer to e^x=4x ?

The answer to e^x=4x is x=-\W_{-1}(-1/4),x=-\W_{0}(-1/4)

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e^x=4x

e^{x} = 4 x

x = - W_{- 1} (- \frac{1}{4}), x = - W_{0} (- \frac{1}{4})

Decimal Notation

x = 2.15329 \dots, x = 0.35740 \dots

Solution steps

e^{x} = 4 x

Prepare

e^{x} = 4 x

for Lambert form:

1 = 4 x e^{- x}

Rewrite the equation with

- x = u

and

x = - u

1 = 4 (- u) e^{u}

Rewrite

1 = 4 (- u) e^{u}

in Lambert form:

e^{u} u = - \frac{1}{4}

Solve

e^{u} u = - \frac{1}{4} : u = W_{- 1} (- \frac{1}{4}), u = W_{0} (- \frac{1}{4})

u = W_{- 1} (- \frac{1}{4}), u = W_{0} (- \frac{1}{4})

Substitute back

u = - x,

solve for

x

Solve

- x = W_{- 1} (- \frac{1}{4}) : x = - W_{- 1} (- \frac{1}{4})

Solve

- x = W_{0} (- \frac{1}{4}) : x = - W_{0} (- \frac{1}{4})

x = - W_{- 1} (- \frac{1}{4}), x = - W_{0} (- \frac{1}{4})

e^{x} = 72

2^{x} = \frac{4}{2}

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

5 e^{4 x + 7} = 15 π

2 7^{2 x - 1} = 24 3^{2 x - 3}

What is the answer to e^x=4x ?
The answer to e^x=4x is x=-\W_{-1}(-1/4),x=-\W_{0}(-1/4)