What is the answer to x^{sqrt(x)}=10 ?

The answer to x^{sqrt(x)}=10 is x=(ln^2(10)}{4\W_{0}(\frac{ln(10))/2)^2}

Solve for x: x^{sqrt(x)}=10

The solution is x=(ln^2(10)}{4\W_{0}(\frac{ln(10))/2)^2}

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x^{x} = 10

x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

Decimal Notation

x = 3.45276 \dots

Solution steps

x^{x} = 10

Prepare

x^{x} = 10

for Lambert form:

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

Rewrite the equation with

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

and

x = \frac{ln ^{2} ( 10 )}{u ^{2}}

(\frac{ln ^{2} ( 10 )}{u ^{2}}) e^{- u} = 1

Solve

(\frac{ln ^{2} ( 10 )}{u ^{2}}) e^{- u} = 1 : u = 2 W_{0} (\frac{ln ( 10 )}{2})

u = 2 W_{0} (\frac{ln ( 10 )}{2})

Substitute back

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

solve for

x

Solve

\frac{ln ( 10 )}{x} = 2 W_{0} (\frac{ln ( 10 )}{2}) : x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

Verify Solutions:

x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

True

The solution is

x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

Verify Solutions:

x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

True

The solution is

x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

5 \cdot 6^{x} = 12 \cdot 3^{x}

3 (5^{x}) - 200 = 175

2^{4 x} = 2^{3 x - 9}

4^{3 x + 5} = \frac{1}{256}

e^{2 x} + 3 e^{x} + 2 = 0

What is the answer to x^{sqrt(x)}=10 ?
The answer to x^{sqrt(x)}=10 is x=(ln^2(10)}{4\W_{0}(\frac{ln(10))/2)^2}
Solve for x: x^{sqrt(x)}=10
The solution is x=(ln^2(10)}{4\W_{0}(\frac{ln(10))/2)^2}