What is the answer to e^x=5-2x ?

The answer to e^x=5-2x is x=-(2\W_{0}((e^{5/2})/2)-5)/(2)

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e^x=5-2x

e^{x} = 5 - 2 x

x = - \frac{2 W _{0} ( \frac{e ^{\frac{5}{2}}}{2} ) - 5}{2}

Decimal Notation

x = 1.05869 \dots

Solution steps

e^{x} = 5 - 2 x

Prepare

e^{x} = 5 - 2 x

for Lambert form:

1 = (5 - 2 x) e^{- x}

Rewrite the equation with

- x + \frac{5}{2} = u

and

x = - \frac{2 u - 5}{2}

1 = (5 - 2 (- \frac{2 u - 5}{2})) e^{- (- \frac{2 u - 5}{2})}

Simplify

1 = 2 e^{\frac{2 u - 5}{2}} u

Rewrite

1 = 2 e^{\frac{2 u - 5}{2}} u

in Lambert form:

e^{u} u = \frac{e ^{\frac{5}{2}}}{2}

Solve

e^{u} u = \frac{e ^{\frac{5}{2}}}{2} : u = W_{0} (\frac{e ^{\frac{5}{2}}}{2})

u = W_{0} (\frac{e ^{\frac{5}{2}}}{2})

Substitute back

u = - x + \frac{5}{2},

solve for

x

Solve

- x + \frac{5}{2} = W_{0} (\frac{e ^{\frac{5}{2}}}{2}) : x = - \frac{2 W _{0} ( \frac{e ^{\frac{5}{2}}}{2} ) - 5}{2}

x = - \frac{2 W _{0} ( \frac{e ^{\frac{5}{2}}}{2} ) - 5}{2}

e^{2 x + 2} + 4 (e^{x + 2} - 1) = e^{x}

3600 = (0.78)^{x} 14500

3^{x} - 6^{3 x + 1} = 0

1200000 = 600000 e^{0.05 t}

9^{x} = 2 7^{x - 2}

What is the answer to e^x=5-2x ?
The answer to e^x=5-2x is x=-(2\W_{0}((e^{5/2})/2)-5)/(2)