What is the answer to 0.1x+e^{0.25x}=2 ?

The answer to 0.1x+e^{0.25x}=2 is x=-(0.1\W_{0}(\frac{0.25e^5)/(0.1))-0.5}{0.025}

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0.1x+e^{0.25x}=2

0.1 x + e^{0.25 x} = 2

x = - \frac{0.1 W _{0} ( \frac{0.25 e ^{5}}{0.1} ) - 0.5}{0.025}

Decimal Notation

x = 2.28688 \dots

Solution steps

0.1 x + e^{0.25 x} = 2

Prepare

0.1 x + e^{0.25 x} = 2

for Lambert form:

1 = (2 - 0.1 x) e^{- 0.25 x}

Rewrite the equation with

- 0.25 x + \frac{0.5}{0.1} = u

and

x = - \frac{u - 5}{0.25}

1 = (2 - 0.1 (- \frac{u - 5}{0.25})) e^{- 0.25 (- \frac{u - 5}{0.25})}

Simplify

1 = e^{u - 5} (2 + \frac{0.1 ( u - 5 )}{0.25})

Rewrite

1 = e^{u - 5} (2 + \frac{0.1 ( u - 5 )}{0.25})

in Lambert form:

e^{u} u = \frac{0.25 e ^{5}}{0.1}

Solve

e^{u} u = \frac{0.25 e ^{5}}{0.1} : u = W_{0} (\frac{0.25 e ^{5}}{0.1})

u = W_{0} (\frac{0.25 e ^{5}}{0.1})

Substitute back

u = - 0.25 x + \frac{0.5}{0.1},

solve for

x

Solve

- 0.25 x + \frac{0.5}{0.1} = W_{0} (\frac{0.25 e ^{5}}{0.1}) : x = - \frac{0.1 W _{0} ( \frac{0.25 e ^{5}}{0.1} ) - 0.5}{0.025}

x = - \frac{0.1 W _{0} ( \frac{0.25 e ^{5}}{0.1} ) - 0.5}{0.025}

2 = 2^{y}

e^{6} x = 6 e^{x}

\frac{4 ^{x + 1}}{2 ^{3 x - 2}} = 256

x, 6^{x} = 2 \cdot y

x, 2^{x + 1} = 5^{1 - 2 x}

What is the answer to 0.1x+e^{0.25x}=2 ?
The answer to 0.1x+e^{0.25x}=2 is x=-(0.1\W_{0}(\frac{0.25e^5)/(0.1))-0.5}{0.025}