What is the answer to e^{2x}+3x=4 ?

The answer to e^{2x}+3x=4 is x=-(3\W_{0}((2e^{8/3})/3)-8)/(6)

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e^{2x}+3x=4

e^{2 x} + 3 x = 4

x = - \frac{3 W _{0} ( \frac{2 e ^{\frac{8}{3}}}{3} ) - 8}{6}

Solution steps

e^{2 x} + 3 x = 4

Prepare

e^{2 x} + 3 x = 4

for Lambert form:

1 = (4 - 3 x) e^{- 2 x}

Rewrite the equation with

- 2 x + \frac{8}{3} = u

and

x = - \frac{3 u - 8}{6}

1 = (4 - 3 (- \frac{3 u - 8}{6})) e^{- 2 (- \frac{3 u - 8}{6})}

Simplify

1 = e^{\frac{3 u - 8}{3}} (4 + \frac{3 u - 8}{2})

Rewrite

1 = e^{\frac{3 u - 8}{3}} (4 + \frac{3 u - 8}{2})

in Lambert form:

e^{u} u = \frac{2 e ^{\frac{8}{3}}}{3}

Solve

e^{u} u = \frac{2 e ^{\frac{8}{3}}}{3} : u = W_{0} (\frac{2 e ^{\frac{8}{3}}}{3})

u = W_{0} (\frac{2 e ^{\frac{8}{3}}}{3})

Substitute back

u = - 2 x + \frac{8}{3},

solve for

x

Solve

- 2 x + \frac{8}{3} = W_{0} (\frac{2 e ^{\frac{8}{3}}}{3}) : x = - \frac{3 W _{0} ( \frac{2 e ^{\frac{8}{3}}}{3} ) - 8}{6}

x = - \frac{3 W _{0} ( \frac{2 e ^{\frac{8}{3}}}{3} ) - 8}{6}

1 6^{x + 3} = 6 4^{2 x - 5}

2500 = 5000 e^{- 0.8 t}

3 e^{4 t} = 200

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

4 e^{4 - x} = 5

What is the answer to e^{2x}+3x=4 ?
The answer to e^{2x}+3x=4 is x=-(3\W_{0}((2e^{8/3})/3)-8)/(6)