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

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

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roots e^x-3x=0

roots

e^{x} - 3 x = 0

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

Solution steps

e^{x} - 3 x = 0

Prepare

e^{x} - 3 x = 0

for Lambert form:

1 = 3 x e^{- x}

Rewrite the equation with

- x = u

and

x = - 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 = - x,

solve for

x

Solve

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

Solve

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

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

y, x = \frac{e ^{y} - e ^{- y}}{2}

6^{x + 2} + 6^{x} = 222

e^{3 x + 2} = 9

2^{4 x + 2} - 48 \cdot 2^{2 x - 2} + 8 = 0

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