What is the answer to e^x=2x+2 ?

The answer to e^x=2x+2 is x=-\W_{-1}(-1/(2e))-1,x=-\W_{0}(-1/(2e))-1

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e^x=2x+2

e^{x} = 2 x + 2

x = - W_{- 1} (- \frac{1}{2 e}) - 1, x = - W_{0} (- \frac{1}{2 e}) - 1

Decimal Notation

x = 1.67834 \dots, x = - 0.76803 \dots

Solution steps

e^{x} = 2 x + 2

Prepare

e^{x} = 2 x + 2

for Lambert form:

1 = (2 x + 2) e^{- x}

Rewrite the equation with

- x - 1 = u

and

x = - u - 1

1 = (2 (- u - 1) + 2) e^{- (- u - 1)}

Simplify

1 = - 2 e^{u + 1} u

Rewrite

1 = - 2 e^{u + 1} u

in Lambert form:

e^{u} u = - \frac{1}{2 e}

Solve

e^{u} u = - \frac{1}{2 e} : u = W_{- 1} (- \frac{1}{2 e}), u = W_{0} (- \frac{1}{2 e})

u = W_{- 1} (- \frac{1}{2 e}), u = W_{0} (- \frac{1}{2 e})

Substitute back

u = - x - 1,

solve for

x

Solve

- x - 1 = W_{- 1} (- \frac{1}{2 e}) : x = - W_{- 1} (- \frac{1}{2 e}) - 1

Solve

- x - 1 = W_{0} (- \frac{1}{2 e}) : x = - W_{0} (- \frac{1}{2 e}) - 1

x = - W_{- 1} (- \frac{1}{2 e}) - 1, x = - W_{0} (- \frac{1}{2 e}) - 1

2^{3 x - 4} = 18

3 \cdot (2)^{x - 1} - 2 = 0

6^{2 x} + 6^{x} - 6 = 0

9 (3^{2 x}) - 22 (3^{x}) - 15 = 0

1 6^{3 x - 6} = 6 4^{x + 4}

What is the answer to e^x=2x+2 ?
The answer to e^x=2x+2 is x=-\W_{-1}(-1/(2e))-1,x=-\W_{0}(-1/(2e))-1