What is the answer to n^{sqrt(n)}=6 ?

The answer to n^{sqrt(n)}=6 is n=(ln^2(6)}{4\W_{0}(\frac{ln(6))/2)^2}

Solve for n: n^{sqrt(n)}=6

The solution is n=(ln^2(6)}{4\W_{0}(\frac{ln(6))/2)^2}

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n^{n} = 6

n = \frac{ln ^{2} ( 6 )}{4 W _{0} ( \frac{l n ( 6 )}{2} ) ^{2}}

Decimal Notation

n = 2.87625 \dots

Solution steps

n^{n} = 6

Prepare

n^{n} = 6

for Lambert form:

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

Rewrite the equation with

\frac{ln ( 6 )}{n} = u

and

n = \frac{ln ^{2} ( 6 )}{u ^{2}}

(\frac{ln ^{2} ( 6 )}{u ^{2}}) e^{- u} = 1

Solve

(\frac{ln ^{2} ( 6 )}{u ^{2}}) e^{- u} = 1 : u = 2 W_{0} (\frac{ln ( 6 )}{2})

u = 2 W_{0} (\frac{ln ( 6 )}{2})

Substitute back

u = \frac{ln ( 6 )}{n},

solve for

n

Solve

\frac{ln ( 6 )}{n} = 2 W_{0} (\frac{ln ( 6 )}{2}) : n = \frac{ln ^{2} ( 6 )}{4 W _{0} ( \frac{l n ( 6 )}{2} ) ^{2}}

n = \frac{ln ^{2} ( 6 )}{4 W _{0} ( \frac{l n ( 6 )}{2} ) ^{2}}

Verify Solutions:

n = \frac{ln ^{2} ( 6 )}{4 W _{0} ( \frac{l n ( 6 )}{2} ) ^{2}}

True

The solution is

n = \frac{ln ^{2} ( 6 )}{4 W _{0} ( \frac{l n ( 6 )}{2} ) ^{2}}

Verify Solutions:

n = \frac{ln ^{2} ( 6 )}{4 W _{0} ( \frac{l n ( 6 )}{2} ) ^{2}}

True

The solution is

n = \frac{ln ^{2} ( 6 )}{4 W _{0} ( \frac{l n ( 6 )}{2} ) ^{2}}

5^{x} - 2 + 5^{x} - 1 = \frac{30}{5}

4^{x} + 4^{2 x} = 272

3 - 2^{x} = - 5

\frac{a ^{x} + 1}{a ^{x}} = n

2^{\frac{2}{( l o g _{5} ( x ) )}} = \frac{1}{16}

What is the answer to n^{sqrt(n)}=6 ?
The answer to n^{sqrt(n)}=6 is n=(ln^2(6)}{4\W_{0}(\frac{ln(6))/2)^2}
Solve for n: n^{sqrt(n)}=6
The solution is n=(ln^2(6)}{4\W_{0}(\frac{ln(6))/2)^2}