What is the answer to x^{4x}=2 ?

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

Solve for x: x^{4x}=2

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

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x^{4 x} = 2

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

Decimal Notation

x = 1.16097 \dots

Solution steps

x^{4 x} = 2

Prepare

x^{4 x} = 2

for Lambert form:

x e^{- \frac{l n ( 2 )}{4 x}} = 1

Rewrite the equation with

\frac{ln ( 2 )}{4 x} = u

and

x = \frac{ln ( 2 )}{4 u}

(\frac{ln ( 2 )}{4 u}) e^{- u} = 1

Rewrite

(\frac{ln ( 2 )}{4 u}) e^{- u} = 1

in Lambert form:

e^{u} u = \frac{ln ( 2 )}{4}

Solve

e^{u} u = \frac{ln ( 2 )}{4} : u = W_{0} (\frac{ln ( 2 )}{4})

Verify Solutions:

u = W_{0} (\frac{ln ( 2 )}{4})

True

The solution is

u = W_{0} (\frac{ln ( 2 )}{4})

Substitute back

u = \frac{ln ( 2 )}{4 x},

solve for

x

Solve

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

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

2 (8^{x}) - 1 = 9

a \cdot b^{t} = c \cdot d^{t}

2^{1000} = 4^{x}

69.53 = 14 + 56 e^{- k (5)}

(3)^{3 - x} = 9^{x + 2}

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