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受欢迎的代数 >

x^x=\sqrt[3]{4/9}

解答

x^{x} = 3 \frac{4}{9}

解答

x = \frac{ln ( \frac{4}{9} )}{3 W _{- 1} ( \frac{l n ( \frac{4}{9} )}{3} )}, x = \frac{ln ( \frac{4}{9} )}{3 W _{0} ( \frac{l n ( \frac{4}{9} )}{3} )}

+1

十进制

x = 0.13497 \dots, x = 0.66666 \dots

求解步骤

x^{x} = 3 \frac{4}{9}

将

x^{x} = 3 \frac{4}{9}

调整为朗伯形式:

x e^{- \frac{l n ( \frac{4}{9} )}{3 x}} = 1

用

\frac{ln ( \frac{4}{9} )}{3 x} = u

和

x = \frac{ln ( \frac{4}{9} )}{3 u}

改写方程式

(\frac{ln ( \frac{4}{9} )}{3 u}) e^{- u} = 1

改写

(\frac{ln ( \frac{4}{9} )}{3 u}) e^{- u} = 1

为朗伯形式:

e^{u} u = \frac{ln ( \frac{4}{9} )}{3}

解

e^{u} u = \frac{ln ( \frac{4}{9} )}{3} : u = W_{- 1} (\frac{ln ( \frac{4}{9} )}{3}), u = W_{0} (\frac{ln ( \frac{4}{9} )}{3})

验证解:

u = W_{- 1} (\frac{ln ( \frac{4}{9} )}{3})

真

, u = W_{0} (\frac{ln ( \frac{4}{9} )}{3})

真

解为

u = W_{- 1} (\frac{ln ( \frac{4}{9} )}{3}), u = W_{0} (\frac{ln ( \frac{4}{9} )}{3})

代回

u = \frac{ln ( \frac{4}{9} )}{3 x}

，求解

x

解

\frac{ln ( \frac{4}{9} )}{3 x} = W_{- 1} (\frac{ln ( \frac{4}{9} )}{3}) : x = \frac{ln ( \frac{4}{9} )}{3 W _{- 1} ( \frac{l n ( \frac{4}{9} )}{3} )}

解

\frac{ln ( \frac{4}{9} )}{3 x} = W_{0} (\frac{ln ( \frac{4}{9} )}{3}) : x = \frac{ln ( \frac{4}{9} )}{3 W _{0} ( \frac{l n ( \frac{4}{9} )}{3} )}

x = \frac{ln ( \frac{4}{9} )}{3 W _{- 1} ( \frac{l n ( \frac{4}{9} )}{3} )}, x = \frac{ln ( \frac{4}{9} )}{3 W _{0} ( \frac{l n ( \frac{4}{9} )}{3} )}

作图

流行的例子

1 = e^{2 x - 3} - 5

e^{2 x + 3} = 27

2^{x - 2} = 8

59 e^{x} = 55

solvefor t,x=e^t+e^{-t}

so l v e f or t, x = e^{t} + e^{- t}