What is the answer to e^x-x-2=0 ?

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

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e^x-x-2=0

e^{x} - x - 2 = 0

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

Decimal Notation

x = 1.14619 \dots, x = - 1.84140 \dots

Solution steps

e^{x} - x - 2 = 0

Prepare

e^{x} - x - 2 = 0

for Lambert form:

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

Rewrite the equation with

- x - 2 = u

and

x = - u - 2

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

Simplify

1 = - e^{u + 2} u

Rewrite

1 = - e^{u + 2} u

in Lambert form:

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

Solve

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

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

Substitute back

u = - x - 2,

solve for

x

Solve

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

Solve

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

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

5^{x - 9} = 2 5^{3 - x}

6^{x} = 7

e^{4 x} = 0

e^{4 x} = 6 e^{x}

3 \cdot 4^{x} + 6 \cdot 2^{x} = 24

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