What is the answer to e^{-x}+x/5-1=0 ?

The answer to e^{-x}+x/5-1=0 is x=0,x=\W_{0}(-5/(e^5))+5

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e^{-x}+x/5-1=0

e^{- x} + \frac{x}{5} - 1 = 0

x = 0, x = W_{0} (- \frac{5}{e ^{5}}) + 5

Decimal Notation

x = 0, x = 4.96511 \dots

Solution steps

e^{- x} + \frac{x}{5} - 1 = 0

Prepare

e^{- x} + \frac{x}{5} - 1 = 0

for Lambert form:

5 = - e^{x} (x - 5)

Rewrite the equation with

x - 5 = u

and

x = u + 5

5 = - e^{(u + 5)} ((u + 5) - 5)

Simplify

5 = - e^{u + 5} u

Rewrite

5 = - e^{u + 5} u

in Lambert form:

e^{u} u = - \frac{5}{e ^{5}}

Solve

e^{u} u = - \frac{5}{e ^{5}} : u = - 5, u = W_{0} (- \frac{5}{e ^{5}})

u = - 5, u = W_{0} (- \frac{5}{e ^{5}})

Substitute back

u = x - 5,

solve for

x

Solve

x - 5 = - 5 : x = 0

Solve

x - 5 = W_{0} (- \frac{5}{e ^{5}}) : x = W_{0} (- \frac{5}{e ^{5}}) + 5

x = 0, x = W_{0} (- \frac{5}{e ^{5}}) + 5

4 9^{4^{x}} = 7

e^{x + 5} = 7

(5^{3})^{n} = 5^{18}

2^{x} - 2^{- x} = 1

1 6^{x} = 2

What is the answer to e^{-x}+x/5-1=0 ?
The answer to e^{-x}+x/5-1=0 is x=0,x=\W_{0}(-5/(e^5))+5