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受欢迎的代数 >

x^x=64

解答

x^{x} = 64

解答

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

+1

十进制

x = 3.39912 \dots

求解步骤

x^{x} = 64

将

x^{x} = 64

调整为朗伯形式:

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

用

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

和

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

改写方程式

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

改写

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

为朗伯形式:

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

解

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

验证解:

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

真

解是

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

代回

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

，求解

x

解

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

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

作图

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