What is the answer to x^x=1000 ?

The answer to x^x=1000 is x=(3ln(10))/(\W_{0)(3ln(10))}

Solve for x: x^x=1000

The solution is x=(3ln(10))/(\W_{0)(3ln(10))}

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x^{x} = 1000

x = \frac{3 ln ( 10 )}{W _{0} ( 3 ln ( 10 ) )}

Decimal

x = 4.55553 \dots

Solution steps

x^{x} = 1000

Prepare

x^{x} = 1000

for Lambert form:

x e^{- \frac{3 l n ( 10 )}{x}} = 1

Rewrite the equation with

\frac{3 ln ( 10 )}{x} = u

and

x = \frac{3 ln ( 10 )}{u}

(\frac{3 ln ( 10 )}{u}) e^{- u} = 1

Rewrite

(\frac{3 ln ( 10 )}{u}) e^{- u} = 1

in Lambert form:

u e^{u} = 3 ln (10)

Solve

u e^{u} = 3 ln (10) : u = W_{0} (3 ln (10))

Verify Solutions:

u = W_{0} (3 ln (10))

True

The solution is

u = W_{0} (3 ln (10))

Substitute back

u = \frac{3 ln ( 10 )}{x},

solve for

x

Solve

\frac{3 ln ( 10 )}{x} = W_{0} (3 ln (10)) : x = \frac{3 ln ( 10 )}{W _{0} ( 3 ln ( 10 ) )}

x = \frac{3 ln ( 10 )}{W _{0} ( 3 ln ( 10 ) )}

y, x = 1 0^{y}

(a^{- x})^{2} = a^{14}

4^{x} - 10 \cdot 4^{- x} = 3

3^{x + 2} = 3^{3 x - 8}

4^{l o g_{4} (x + 8)} = 4^{2}

What is the answer to x^x=1000 ?
The answer to x^x=1000 is x=(3ln(10))/(\W_{0)(3ln(10))}
Solve for x: x^x=1000
The solution is x=(3ln(10))/(\W_{0)(3ln(10))}