What is the answer to e^x=3x+5 ?

The answer to e^x=3x+5 is x=-(3\W_{-1}(-1/(3e^{5/3)})+5)/3 ,x=-(3\W_{0}(-1/(3e^{5/3)})+5)/3

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e^x=3x+5

e^{x} = 3 x + 5

x = - \frac{3 W _{- 1} ( - \frac{1}{3 e ^{\frac{5}{3}}} ) + 5}{3}, x = - \frac{3 W _{0} ( - \frac{1}{3 e ^{\frac{5}{3}}} ) + 5}{3}

Solution steps

e^{x} = 3 x + 5

Prepare

e^{x} = 3 x + 5

for Lambert form:

1 = (3 x + 5) e^{- x}

Rewrite the equation with

- x - \frac{5}{3} = u

and

x = - \frac{3 u + 5}{3}

1 = (3 (- \frac{3 u + 5}{3}) + 5) e^{- (- \frac{3 u + 5}{3})}

Simplify

1 = - 3 e^{\frac{3 u + 5}{3}} u

Rewrite

1 = - 3 e^{\frac{3 u + 5}{3}} u

in Lambert form:

e^{u} u = - \frac{1}{3 e ^{\frac{5}{3}}}

Solve

e^{u} u = - \frac{1}{3 e ^{\frac{5}{3}}} : u = W_{- 1} (- \frac{1}{3 e ^{\frac{5}{3}}}), u = W_{0} (- \frac{1}{3 e ^{\frac{5}{3}}})

u = W_{- 1} (- \frac{1}{3 e ^{\frac{5}{3}}}), u = W_{0} (- \frac{1}{3 e ^{\frac{5}{3}}})

Substitute back

u = - x - \frac{5}{3},

solve for

x

Solve

- x - \frac{5}{3} = W_{- 1} (- \frac{1}{3 e ^{\frac{5}{3}}}) : x = - \frac{3 W _{- 1} ( - \frac{1}{3 e ^{\frac{5}{3}}} ) + 5}{3}

Solve

- x - \frac{5}{3} = W_{0} (- \frac{1}{3 e ^{\frac{5}{3}}}) : x = - \frac{3 W _{0} ( - \frac{1}{3 e ^{\frac{5}{3}}} ) + 5}{3}

x = - \frac{3 W _{- 1} ( - \frac{1}{3 e ^{\frac{5}{3}}} ) + 5}{3}, x = - \frac{3 W _{0} ( - \frac{1}{3 e ^{\frac{5}{3}}} ) + 5}{3}

1000 e^{0.2 x} = 2500

8^{- 2 - x} = 431

6^{x + 3} = 3^{x}

0.5 \cdot 1 0^{8 t} = 73

4^{x} = 6^{1 - x}

What is the answer to e^x=3x+5 ?
The answer to e^x=3x+5 is x=-(3\W_{-1}(-1/(3e^{5/3)})+5)/3 ,x=-(3\W_{0}(-1/(3e^{5/3)})+5)/3