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受欢迎的代数 >

(1/2)^x=x

解答

(\frac{1}{2})^{x} = x

解答

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

+1

十进制

x = 0.64118 \dots

求解步骤

(\frac{1}{2})^{x} = x

将

(\frac{1}{2})^{x} = x

调整为朗伯形式:

1 = x e^{l n (2) x}

用

ln (2) x = u

和

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

改写方程式

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

改写

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

为朗伯形式:

e^{u} u = ln (2)

解

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

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

代回

u = ln (2) x

，求解

x

解

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

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

作图

流行的例子

4^{x - 1} = 16

2^{2x}+2^{x+1}=80

2^{2 x} + 2^{x + 1} = 80

8^{y} = 32

3^{x} = 15

e^{2 x} = 6 e^{x}