What is the answer to solvefor x,x^x=y^y ?

The answer to solvefor x,x^x=y^y is x=(yln(y))/(\W(yln(y)))

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solvefor x,x^x=y^y

solve for

x, x^{x} = y^{y}

x = \frac{y ln ( y )}{W ( y ln ( y ) )}

Solution steps

x^{x} = y^{y}

Prepare

x^{x} = y^{y}

for Lambert form:

x e^{- \frac{y l n ( y )}{x}} = 1

Rewrite the equation with

\frac{y ln ( y )}{x} = u

and

x = \frac{y ln ( y )}{u}

\frac{y ln ( y )}{u} e^{- u} = 1

Rewrite

(\frac{y ln ( y )}{u}) e^{- u} = 1

in Lambert form:

u e^{u} = y ln (y)

Solve

u e^{u} = y ln (y) : u = W (y ln (y))

Verify Solutions:

u = W (y ln (y))

True

The solution is

u = W (y ln (y))

Substitute back

u = \frac{y ln ( y )}{x},

solve for

x

Solve

\frac{y ln ( y )}{x} = W (y ln (y)) : x = \frac{y ln ( y )}{W ( y ln ( y ) )}

x = \frac{y ln ( y )}{W ( y ln ( y ) )}

1 6^{2 + x + 1} = 64

- 3 e^{3 x} x - e^{3 x} = 0

3^{2 x} - 2 \cdot 3^{x + 5} + 3^{10} = 0

1 7^{4 - 3 x} = 1

7 = 240 1^{x}

What is the answer to solvefor x,x^x=y^y ?
The answer to solvefor x,x^x=y^y is x=(yln(y))/(\W(yln(y)))