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受欢迎的代数 >

x^x=\sqrt[9]{1/3}

解答

x^{x} = 9 \frac{1}{3}

解答

x = \frac{1}{27}, x = - \frac{ln ( 3 )}{9 W _{0} ( - \frac{l n ( 3 )}{9} )}

+1

十进制

x = 0.03703 \dots, x = 0.86894 \dots

求解步骤

x^{x} = 9 \frac{1}{3}

将

x^{x} = 9 \frac{1}{3}

调整为朗伯形式:

x e^{\frac{l n ( 3 )}{9 x}} = 1

用

- \frac{ln ( 3 )}{9 x} = u

和

x = - \frac{ln ( 3 )}{9 u}

改写方程式

(- \frac{ln ( 3 )}{9 u}) e^{- u} = 1

解

(- \frac{ln ( 3 )}{9 u}) e^{- u} = 1 : u = - 3 ln (3), u = W_{0} (- \frac{ln ( 3 )}{9})

u = - 3 ln (3), u = W_{0} (- \frac{ln ( 3 )}{9})

代回

u = - \frac{ln ( 3 )}{9 x}

，求解

x

解

- \frac{ln ( 3 )}{9 x} = - 3 ln (3) : x = \frac{1}{27}

解

- \frac{ln ( 3 )}{9 x} = W_{0} (- \frac{ln ( 3 )}{9}) : x = - \frac{ln ( 3 )}{9 W _{0} ( - \frac{l n ( 3 )}{9} )}

x = \frac{1}{27}, x = - \frac{ln ( 3 )}{9 W _{0} ( - \frac{l n ( 3 )}{9} )}

作图

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