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受欢迎的代数 >

x^x=2^{1000000}

解答

x^{x} = 2^{1000000}

解答

x = \frac{1000000 ln ( 2 )}{W _{0} ( 1000000 ln ( 2 ) )}

+1

十进制

x = 62746.12646 \dots

求解步骤

x^{x} = 2^{1000000}

将

x^{x} = 2^{1000000}

调整为朗伯形式:

x e^{- \frac{1000000 l n ( 2 )}{x}} = 1

用

\frac{1000000 ln ( 2 )}{x} = u

和

x = \frac{1000000 ln ( 2 )}{u}

改写方程式

(\frac{1000000 ln ( 2 )}{u}) e^{- u} = 1

改写

(\frac{1000000 ln ( 2 )}{u}) e^{- u} = 1

为朗伯形式:

u e^{u} = 1000000 ln (2)

解

u e^{u} = 1000000 ln (2) : u = W_{0} (1000000 ln (2))

验证解:

u = W_{0} (1000000 ln (2))

真

解是

u = W_{0} (1000000 ln (2))

代回

u = \frac{1000000 ln ( 2 )}{x}

，求解

x

解

\frac{1000000 ln ( 2 )}{x} = W_{0} (1000000 ln (2)) : x = \frac{1000000 ln ( 2 )}{W _{0} ( 1000000 ln ( 2 ) )}

x = \frac{1000000 ln ( 2 )}{W _{0} ( 1000000 ln ( 2 ) )}

作图

流行的例子

216=(1/36)^{x-1}

216 = (\frac{1}{36})^{x - 1}

x^{x} = 256

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