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Popular >

x^e=e^x

Solution

x^{e} = e^{x}

Solution

x = e

+1

Solution steps

x^{e} = e^{x}

Prepare

x^{e} = e^{x}

for Lambert form:

x e^{- \frac{x}{e}} = 1

Rewrite the equation with

- \frac{x}{e} = u

and

x = - e u

(- e u) e^{u} = 1

Rewrite

(- e u) e^{u} = 1

in Lambert form:

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

Solve

e^{u} u = - \frac{1}{e} : u = - 1

u = - 1

Substitute back

u = - \frac{x}{e},

solve for

x

Solve

- \frac{x}{e} = - 1 : x = e

x = e

Verify Solutions:

x = e

True

The solution is

x = e

Verify Solutions:

x = e

True

The solution is

x = e

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a e^{x}

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3^{2(x+1)}-18\times 3^x+9=0

3^{2 (x + 1)} - 18 \times 3^{x} + 9 = 0

6^{3m}*6^{-m}=6^{-2m}

6^{3 m} \cdot 6^{- m} = 6^{- 2 m}

100 = 50 e^{10 x}

Frequently Asked Questions (FAQ)

What is the answer to x^e=e^x ?
The answer to x^e=e^x is x=e