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

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

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0=e^x-3x

0 = e^{x} - 3 x

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

Decimal Notation

x = 1.51213 \dots, x = 0.61906 \dots

Solution steps

0 = e^{x} - 3 x

Prepare

0 = e^{x} - 3 x

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})

4^{x} = (\frac{1}{4})^{x + 3}

30000 = 700 \cdot 2^{2 t}

375 = 1500 \cdot 0.99 3^{t}

1000 = 7 \cdot 1. 1^{n - 1}

3^{2 x + 8} = 9^{3 x - 10}

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