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受欢迎的代数 >

x^x=sqrt(2)

解答

x^{x} = 2

解答

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

+1

十进制

x = 1.30435 \dots

求解步骤

x^{x} = 2

将

x^{x} = 2

调整为朗伯形式:

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

用

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

和

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

改写方程式

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

改写

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

为朗伯形式:

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

解

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

验证解:

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

真

解是

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

代回

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

，求解

x

解

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

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

作图

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