What is the answer to e^x=3-2x ?

The answer to e^x=3-2x is x=-(2\W_{0}((e^{3/2})/2)-3)/(2)

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e^x=3-2x

e^{x} = 3 - 2 x

x = - \frac{2 W _{0} ( \frac{e ^{\frac{3}{2}}}{2} ) - 3}{2}

Decimal Notation

x = 0.59420 \dots

Solution steps

e^{x} = 3 - 2 x

Prepare

e^{x} = 3 - 2 x

for Lambert form:

1 = (3 - 2 x) e^{- x}

Rewrite the equation with

- x + \frac{3}{2} = u

and

x = - \frac{2 u - 3}{2}

1 = (3 - 2 (- \frac{2 u - 3}{2})) e^{- (- \frac{2 u - 3}{2})}

Simplify

1 = 2 e^{\frac{2 u - 3}{2}} u

Rewrite

1 = 2 e^{\frac{2 u - 3}{2}} u

in Lambert form:

e^{u} u = \frac{e ^{\frac{3}{2}}}{2}

Solve

e^{u} u = \frac{e ^{\frac{3}{2}}}{2} : u = W_{0} (\frac{e ^{\frac{3}{2}}}{2})

u = W_{0} (\frac{e ^{\frac{3}{2}}}{2})

Substitute back

u = - x + \frac{3}{2},

solve for

x

Solve

- x + \frac{3}{2} = W_{0} (\frac{e ^{\frac{3}{2}}}{2}) : x = - \frac{2 W _{0} ( \frac{e ^{\frac{3}{2}}}{2} ) - 3}{2}

x = - \frac{2 W _{0} ( \frac{e ^{\frac{3}{2}}}{2} ) - 3}{2}

4^{x} \cdot 2^{x^{2}} = 8^{x^{3}}

7^{x} = 14

1 5^{- 3} \cdot 1 5^{b} = 1 5^{- 5}

5^{x - 1} = 3

x, x y = e^{2 x + y}

What is the answer to e^x=3-2x ?
The answer to e^x=3-2x is x=-(2\W_{0}((e^{3/2})/2)-3)/(2)