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受欢迎的代数 >

x^x=\sqrt[4]{2}

解答

x^{x} = 42

解答

x = \frac{ln ( 2 )}{4 W _{0} ( \frac{l n ( 2 )}{4} )}

+1

十进制

x = 1.16097 \dots

求解步骤

x^{x} = 42

将

x^{x} = 42

调整为朗伯形式:

x e^{- \frac{l n ( 2 )}{4 x}} = 1

用

\frac{ln ( 2 )}{4 x} = u

和

x = \frac{ln ( 2 )}{4 u}

改写方程式

(\frac{ln ( 2 )}{4 u}) e^{- u} = 1

改写

(\frac{ln ( 2 )}{4 u}) e^{- u} = 1

为朗伯形式:

e^{u} u = \frac{ln ( 2 )}{4}

解

e^{u} u = \frac{ln ( 2 )}{4} : u = W_{0} (\frac{ln ( 2 )}{4})

验证解:

u = W_{0} (\frac{ln ( 2 )}{4})

真

解是

u = W_{0} (\frac{ln ( 2 )}{4})

代回

u = \frac{ln ( 2 )}{4 x}

，求解

x

解

\frac{ln ( 2 )}{4 x} = W_{0} (\frac{ln ( 2 )}{4}) : x = \frac{ln ( 2 )}{4 W _{0} ( \frac{l n ( 2 )}{4} )}

x = \frac{ln ( 2 )}{4 W _{0} ( \frac{l n ( 2 )}{4} )}

作图

流行的例子

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5^{3 x - 1} = (\frac{1}{25})^{2 x + 3}

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