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受欢迎的代数 >

x^{sqrt(x)}=10

解答

x^{x} = 10

解答

x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

+1

十进制

x = 3.45276 \dots

求解步骤

x^{x} = 10

将

x^{x} = 10

调整为朗伯形式:

x e^{- \frac{l n ( 10 )}{x}} = 1

用

\frac{ln ( 10 )}{x} = u

和

x = \frac{ln ^{2} ( 10 )}{u ^{2}}

改写方程式

(\frac{ln ^{2} ( 10 )}{u ^{2}}) e^{- u} = 1

解

(\frac{ln ^{2} ( 10 )}{u ^{2}}) e^{- u} = 1 : u = 2 W_{0} (\frac{ln ( 10 )}{2})

u = 2 W_{0} (\frac{ln ( 10 )}{2})

代回

u = \frac{ln ( 10 )}{x}

，求解

x

解

\frac{ln ( 10 )}{x} = 2 W_{0} (\frac{ln ( 10 )}{2}) : x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

验证解:

x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

真

解是

x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

验证解:

x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

真

解是

x = \frac{ln ^{2} ( 10 )}{4 W _{0} ( \frac{l n ( 10 )}{2} ) ^{2}}

作图

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