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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

solvefor y,x^x=y^y

solve for

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

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

Solution steps

x^{x} = y^{y}

Prepare

x^{x} = y^{y}

for Lambert form:

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

Rewrite the equation with

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

and

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

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

Rewrite

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

in Lambert form:

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

Solve

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

Verify Solutions:

u = W (x ln (x))

True

The solution is

u = W (x ln (x))

Substitute back

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

solve for

y

Solve

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

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

(\frac{1}{2})^{x} = 128

5 e^{x} - 12 e^{- x} - 11 = 0

7^{3 x - 1} = 5^{x + 1}

2^{2 x - 4} = 8^{4}

1 6^{2 x - 2} = 2^{3 x + 3}

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