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受欢迎的代数 >

5^x=x^5

解答

5^{x} = x^{5}

解答

x = 5, x = - \frac{5 W _{0} ( - \frac{l n ( 5 )}{5} )}{ln ( 5 )}

+1

十进制

x = 5, x = 1.76492 \dots

求解步骤

5^{x} = x^{5}

将

5^{x} = x^{5}

调整为朗伯形式:

1 = x e^{- \frac{l n ( 5 ) x}{5}}

用

- \frac{ln ( 5 ) x}{5} = u

和

x = - \frac{5 u}{ln ( 5 )}

改写方程式

1 = (- \frac{5 u}{ln ( 5 )}) e^{u}

改写

1 = (- \frac{5 u}{ln ( 5 )}) e^{u}

为朗伯形式:

e^{u} u = - \frac{ln ( 5 )}{5}

解

e^{u} u = - \frac{ln ( 5 )}{5} : u = - ln (5), u = W_{0} (- \frac{ln ( 5 )}{5})

u = - ln (5), u = W_{0} (- \frac{ln ( 5 )}{5})

代回

u = - \frac{ln ( 5 ) x}{5}

，求解

x

解

- \frac{ln ( 5 ) x}{5} = - ln (5) : x = 5

解

- \frac{ln ( 5 ) x}{5} = W_{0} (- \frac{ln ( 5 )}{5}) : x = - \frac{5 W _{0} ( - \frac{l n ( 5 )}{5} )}{ln ( 5 )}

x = 5, x = - \frac{5 W _{0} ( - \frac{l n ( 5 )}{5} )}{ln ( 5 )}

作图

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