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受欢迎的代数 >

x^5=5^x

解答

x^{5} = 5^{x}

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

x^{5} = 5^{x}

将

x^{5} = 5^{x}

调整为朗伯形式:

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

用

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

和

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

改写方程式

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

改写

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

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

流行的例子

4^{x + 1} = 64

2^{2 x + 1} = 4

e^{x} = - 3

(1/4)^{4x}=0.25

(\frac{1}{4})^{4 x} = 0.25

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