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Popular >

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

Solution

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

Solution

x = 0.13665 \dots

Solution steps

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

Prepare

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

for Lambert form:

1 = (2.3 x + 1) e^{- 2 x}

Rewrite the equation with

- 2 x - \frac{2}{2.3} = u

and

x = - \frac{u + 0.86956 \dots}{2}

1 = (2.3 (- \frac{u + 0.86956 \dots}{2}) + 1) e^{- 2 (- \frac{u + 0.86956 \dots}{2})}

Simplify

1 = e^{u + 0.86956 \dots} (- \frac{2.3 ( u + 0.86956 \dots )}{2} + 1)

Rewrite

1 = e^{u + 0.86956 \dots} (- \frac{2.3 ( u + 0.86956 \dots )}{2} + 1)

in Lambert form:

e^{u} u = - \frac{2}{2.3 e ^{0.86956 \dots}}

Solve

e^{u} u = - \frac{2}{2.3 e ^{0.86956 \dots}} : u = W_{- 1} (- \frac{2}{2.3 e ^{0.86956 \dots}}), u = W_{0} (- \frac{2}{2.3 e ^{0.86956 \dots}})

u = W_{- 1} (- \frac{2}{2.3 e ^{0.86956 \dots}}), u = W_{0} (- \frac{2}{2.3 e ^{0.86956 \dots}})

Substitute back

u = - 2 x - \frac{2}{2.3},

solve for

x

Solve

- 2 x - \frac{2}{2.3} = W_{- 1} (- \frac{2}{2.3 e ^{0.86956 \dots}}) : x = 0.13665 \dots

Solve

- 2 x - \frac{2}{2.3} = W_{0} (- \frac{2}{2.3 e ^{0.86956 \dots}}) : x = - \frac{2.3 W _{0} ( - \frac{2}{2.3 e ^{0.86956 \dots}} ) + 2}{4.6}

x = 0.13665 \dots, x = - \frac{2.3 W _{0} ( - \frac{2}{2.3 e ^{0.86956 \dots}} ) + 2}{4.6}

Verify Solutions:

x = 0.13665 \dots

True

, x = - \frac{2.3 W _{0} ( - \frac{2}{2.3 e ^{0.86956 \dots}} ) + 2}{4.6}

False

The solution is

x = 0.13665 \dots

Verify Solutions:

x = 0.13665 \dots

True

The solution is

x = 0.13665 \dots

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3^{2 x} = 7^{2 x - 1}

1.0075^x=1.251445278

1.007 5^{x} = 1.251445278

Frequently Asked Questions (FAQ)

What is the answer to 2.3=((e^{2x}-1))/x ?
The answer to 2.3=((e^{2x}-1))/x is x=0.13665…