What is the answer to x^x=2^{10^6} ?

The answer to x^x=2^{10^6} is x=(1000000ln(2))/(\W_{0)(1000000ln(2))}

Solve for x: x^x=2^{10^6}

The solution is x=(1000000ln(2))/(\W_{0)(1000000ln(2))}

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x^{x} = 2^{1 0^{6}}

x = \frac{1000000 ln ( 2 )}{W _{0} ( 1000000 ln ( 2 ) )}

Solution steps

x^{x} = 2^{1 0^{6}}

Prepare

x^{x} = 2^{1 0^{6}}

for Lambert form:

x e^{- \frac{1000000 l n ( 2 )}{x}} = 1

Rewrite the equation with

\frac{1000000 ln ( 2 )}{x} = u

and

x = \frac{1000000 ln ( 2 )}{u}

(\frac{1000000 ln ( 2 )}{u}) e^{- u} = 1

Rewrite

(\frac{1000000 ln ( 2 )}{u}) e^{- u} = 1

in Lambert form:

u e^{u} = 1000000 ln (2)

Solve

u e^{u} = 1000000 ln (2) : u = W_{0} (1000000 ln (2))

Verify Solutions:

u = W_{0} (1000000 ln (2))

True

The solution is

u = W_{0} (1000000 ln (2))

Substitute back

u = \frac{1000000 ln ( 2 )}{x},

solve for

x

Solve

\frac{1000000 ln ( 2 )}{x} = W_{0} (1000000 ln (2)) : x = \frac{1000000 ln ( 2 )}{W _{0} ( 1000000 ln ( 2 ) )}

x = \frac{1000000 ln ( 2 )}{W _{0} ( 1000000 ln ( 2 ) )}

6^{3 x + 4} = 3 6^{2 x - 3}

x, (x y)^{x} = e

2^{2 x + 4} = 3^{x - 24}

2^{2 - x} = 5^{x + 1}

2^{n} = 100

What is the answer to x^x=2^{10^6} ?
The answer to x^x=2^{10^6} is x=(1000000ln(2))/(\W_{0)(1000000ln(2))}
Solve for x: x^x=2^{10^6}
The solution is x=(1000000ln(2))/(\W_{0)(1000000ln(2))}