What is the answer to e^x(x-5)+5=0 ?

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

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e^x(x-5)+5=0

e^{x} (x - 5) + 5 = 0

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

Solution steps

e^{x} (x - 5) + 5 = 0

Prepare

e^{x} (x - 5) + 5 = 0

for Lambert form:

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

Rewrite the equation with

x - 5 = u

and

x = u + 5

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

Simplify

e^{u + 5} u = - 5

Rewrite

e^{u + 5} u = - 5

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

3^{17} = 3^{2 x + 3}

e^{3 x} = e^{- x}

5^{3 x - 4} = 5^{2 x + 5}

2^{x} \cdot 8^{3 x - 1} = 64

6^{x + 1} - 18 (6^{x - 1}) = 108

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