What is the answer to solvefor y,x^y-y^x=0 ?

The answer to solvefor y,x^y-y^x=0 is y=-(\W(-(ln(x))/x)x)/(ln(x))

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solvefor y,x^y-y^x=0

solve for

y, x^{y} - y^{x} = 0

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

Solution steps

x^{y} - y^{x} = 0

Prepare

x^{y} - y^{x} = 0

for Lambert form:

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

Rewrite the equation with

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

and

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

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

Rewrite

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

in Lambert form:

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

Solve

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

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

Substitute back

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

solve for

y

Solve

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

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

64 = 2^{x - 1}

3 \cdot 2^{x} = 1

3^{x + 4} = 243

21 6^{x} = 6^{x + 10}

2^{2 x + 1} = 3 2^{x}

What is the answer to solvefor y,x^y-y^x=0 ?
The answer to solvefor y,x^y-y^x=0 is y=-(\W(-(ln(x))/x)x)/(ln(x))