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

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

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solvefor x,y^x=x^y

solve for

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

x = - \frac{W ( - \frac{l n ( y )}{y} ) y}{ln ( y )}

Solution steps

y^{x} = x^{y}

Prepare

y^{x} = x^{y}

for Lambert form:

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

Rewrite the equation with

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

and

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

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

Rewrite

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

in Lambert form:

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

Solve

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

u = W (- \frac{ln ( y )}{y})

Substitute back

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

solve for

x

Solve

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

x = - \frac{W ( - \frac{l n ( y )}{y} ) y}{ln ( y )}

e^{2 x} = 2 e^{x}

x, 2^{x} + 2^{y} = 18

2^{n} = 60000

2^{5 x - 2} = 4^{x}

0. 2^{x + 1} = 125

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