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Popular >

((e^{2x}-1))/x =2.4

Solution

\frac{( e ^{2 x} - 1 )}{x} = 2.4

Solution

x = 0.17709 \dots

Solution steps

\frac{( e ^{2 x} - 1 )}{x} = 2.4

Prepare

\frac{( e ^{2 x} - 1 )}{x} = 2.4

for Lambert form:

1 = (2.4 x + 1) e^{- 2 x}

Rewrite the equation with

- 2 x - \frac{2}{2.4} = u

and

x = - \frac{u + 0.83333 \dots}{2}

1 = (2.4 (- \frac{u + 0.83333 \dots}{2}) + 1) e^{- 2 (- \frac{u + 0.83333 \dots}{2})}

Simplify

1 = e^{u + 0.83333 \dots} (- \frac{2.4 ( u + 0.83333 \dots )}{2} + 1)

Rewrite

1 = e^{u + 0.83333 \dots} (- \frac{2.4 ( u + 0.83333 \dots )}{2} + 1)

in Lambert form:

e^{u} u = - \frac{2}{2.4 e ^{0.83333 \dots}}

Solve

e^{u} u = - \frac{2}{2.4 e ^{0.83333 \dots}} : u = W_{- 1} (- \frac{2}{2.4 e ^{0.83333 \dots}}), u = W_{0} (- \frac{2}{2.4 e ^{0.83333 \dots}})

u = W_{- 1} (- \frac{2}{2.4 e ^{0.83333 \dots}}), u = W_{0} (- \frac{2}{2.4 e ^{0.83333 \dots}})

Substitute back

u = - 2 x - \frac{2}{2.4},

solve for

x

Solve

- 2 x - \frac{2}{2.4} = W_{- 1} (- \frac{2}{2.4 e ^{0.83333 \dots}}) : x = 0.17709 \dots

Solve

- 2 x - \frac{2}{2.4} = W_{0} (- \frac{2}{2.4 e ^{0.83333 \dots}}) : x = - \frac{2.4 W _{0} ( - \frac{2}{2.4 e ^{0.83333 \dots}} ) + 2}{4.8}

x = 0.17709 \dots, x = - \frac{2.4 W _{0} ( - \frac{2}{2.4 e ^{0.83333 \dots}} ) + 2}{4.8}

Verify Solutions:

x = 0.17709 \dots

True

, x = - \frac{2.4 W _{0} ( - \frac{2}{2.4 e ^{0.83333 \dots}} ) + 2}{4.8}

False

The solution is

x = 0.17709 \dots

Verify Solutions:

x = 0.17709 \dots

True

The solution is

x = 0.17709 \dots

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Frequently Asked Questions (FAQ)

What is the answer to ((e^{2x}-1))/x =2.4 ?
The answer to ((e^{2x}-1))/x =2.4 is x=0.17709…