What is the answer to x^x=10^6 ?

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

Solve for x: x^x=10^6

The solution is x=(6ln(10))/(\W_{0)(6ln(10))}

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

x = \frac{6 ln ( 10 )}{W _{0} ( 6 ln ( 10 ) )}

Decimal Notation

x = 7.06579 \dots

Solution steps

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

Prepare

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

for Lambert form:

x e^{- \frac{6 l n ( 10 )}{x}} = 1

Rewrite the equation with

\frac{6 ln ( 10 )}{x} = u

and

x = \frac{6 ln ( 10 )}{u}

(\frac{6 ln ( 10 )}{u}) e^{- u} = 1

Rewrite

(\frac{6 ln ( 10 )}{u}) e^{- u} = 1

in Lambert form:

u e^{u} = 6 ln (10)

Solve

u e^{u} = 6 ln (10) : u = W_{0} (6 ln (10))

Verify Solutions:

u = W_{0} (6 ln (10))

True

The solution is

u = W_{0} (6 ln (10))

Substitute back

u = \frac{6 ln ( 10 )}{x},

solve for

x

Solve

\frac{6 ln ( 10 )}{x} = W_{0} (6 ln (10)) : x = \frac{6 ln ( 10 )}{W _{0} ( 6 ln ( 10 ) )}

x = \frac{6 ln ( 10 )}{W _{0} ( 6 ln ( 10 ) )}

9^{2 x - 7} = \frac{1}{3}

3 (12)^{- 3 x} - 5 = 41

5 e^{3 x + 2} = 5

x^{2} - 2 x e^{- x} + e^{- 2 x} = 0

3 2^{2 x - 3} = \frac{1}{8}

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