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受欢迎的代数 >

x^x=2^{1/6}

解答

x^{x} = 2^{\frac{1}{6}}

解答

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

+1

十进制

x = 1.10971 \dots

求解步骤

x^{x} = 2^{\frac{1}{6}}

将

x^{x} = 2^{\frac{1}{6}}

调整为朗伯形式:

x e^{- \frac{l n ( 2 )}{6 x}} = 1

用

\frac{ln ( 2 )}{6 x} = u

和

x = \frac{ln ( 2 )}{6 u}

改写方程式

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

改写

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

为朗伯形式:

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

解

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

验证解:

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

真

解是

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

代回

u = \frac{ln ( 2 )}{6 x}

，求解

x

解

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

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

作图

流行的例子

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3 e^{5 x} - 2 = 10

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\frac{a x}{e ^{b x}} = 0

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