What is the answer to e^x-6x=5 ?

The answer to e^x-6x=5 is x=-(6\W_{-1}(-1/(6e^{5/6)})+5)/6 ,x=-(6\W_{0}(-1/(6e^{5/6)})+5)/6

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e^x-6x=5

e^{x} - 6 x = 5

x = - \frac{6 W _{- 1} ( - \frac{1}{6 e ^{\frac{5}{6}}} ) + 5}{6}, x = - \frac{6 W _{0} ( - \frac{1}{6 e ^{\frac{5}{6}}} ) + 5}{6}

Decimal Notation

x = 3.18183 \dots, x = - 0.75499 \dots

Solution steps

e^{x} - 6 x = 5

Prepare

e^{x} - 6 x = 5

for Lambert form:

1 = (5 + 6 x) e^{- x}

Rewrite the equation with

- x - \frac{5}{6} = u

and

x = - \frac{6 u + 5}{6}

1 = (5 + 6 (- \frac{6 u + 5}{6})) e^{- (- \frac{6 u + 5}{6})}

Simplify

1 = - 6 e^{\frac{6 u + 5}{6}} u

Rewrite

1 = - 6 e^{\frac{6 u + 5}{6}} u

in Lambert form:

e^{u} u = - \frac{1}{6 e ^{\frac{5}{6}}}

Solve

e^{u} u = - \frac{1}{6 e ^{\frac{5}{6}}} : u = W_{- 1} (- \frac{1}{6 e ^{\frac{5}{6}}}), u = W_{0} (- \frac{1}{6 e ^{\frac{5}{6}}})

u = W_{- 1} (- \frac{1}{6 e ^{\frac{5}{6}}}), u = W_{0} (- \frac{1}{6 e ^{\frac{5}{6}}})

Substitute back

u = - x - \frac{5}{6},

solve for

x

Solve

- x - \frac{5}{6} = W_{- 1} (- \frac{1}{6 e ^{\frac{5}{6}}}) : x = - \frac{6 W _{- 1} ( - \frac{1}{6 e ^{\frac{5}{6}}} ) + 5}{6}

Solve

- x - \frac{5}{6} = W_{0} (- \frac{1}{6 e ^{\frac{5}{6}}}) : x = - \frac{6 W _{0} ( - \frac{1}{6 e ^{\frac{5}{6}}} ) + 5}{6}

x = - \frac{6 W _{- 1} ( - \frac{1}{6 e ^{\frac{5}{6}}} ) + 5}{6}, x = - \frac{6 W _{0} ( - \frac{1}{6 e ^{\frac{5}{6}}} ) + 5}{6}

2^{x} \cdot 2^{x} = 2^{1}

8^{x - 10} = 0.2 5^{x}

5^{x + 3} 5^{2 x - 4} = 125

3^{3 x - 4} = 12

1 3^{x} = 1

What is the answer to e^x-6x=5 ?
The answer to e^x-6x=5 is x=-(6\W_{-1}(-1/(6e^{5/6)})+5)/6 ,x=-(6\W_{0}(-1/(6e^{5/6)})+5)/6