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受欢迎的代数 >

2^x-x^3=0

解答

2^{x} - x^{3} = 0

解答

x = - \frac{3 W _{- 1} ( - \frac{l n ( 2 )}{3} )}{ln ( 2 )}, x = - \frac{3 W _{0} ( - \frac{l n ( 2 )}{3} )}{ln ( 2 )}

+1

十进制

x = 9.93953 \dots, x = 1.37346 \dots

求解步骤

2^{x} - x^{3} = 0

将

2^{x} - x^{3} = 0

调整为朗伯形式:

1 = x e^{- \frac{l n ( 2 ) x}{3}}

用

- \frac{ln ( 2 ) x}{3} = u

和

x = - \frac{3 u}{ln ( 2 )}

改写方程式

1 = (- \frac{3 u}{ln ( 2 )}) e^{u}

改写

1 = (- \frac{3 u}{ln ( 2 )}) e^{u}

为朗伯形式:

e^{u} u = - \frac{ln ( 2 )}{3}

解

e^{u} u = - \frac{ln ( 2 )}{3} : u = W_{- 1} (- \frac{ln ( 2 )}{3}), u = W_{0} (- \frac{ln ( 2 )}{3})

u = W_{- 1} (- \frac{ln ( 2 )}{3}), u = W_{0} (- \frac{ln ( 2 )}{3})

代回

u = - \frac{ln ( 2 ) x}{3}

，求解

x

解

- \frac{ln ( 2 ) x}{3} = W_{- 1} (- \frac{ln ( 2 )}{3}) : x = - \frac{3 W _{- 1} ( - \frac{l n ( 2 )}{3} )}{ln ( 2 )}

解

- \frac{ln ( 2 ) x}{3} = W_{0} (- \frac{ln ( 2 )}{3}) : x = - \frac{3 W _{0} ( - \frac{l n ( 2 )}{3} )}{ln ( 2 )}

x = - \frac{3 W _{- 1} ( - \frac{l n ( 2 )}{3} )}{ln ( 2 )}, x = - \frac{3 W _{0} ( - \frac{l n ( 2 )}{3} )}{ln ( 2 )}

作图

流行的例子

6 (5^{x - 4}) = 42

9^{x} = 2 7^{3 - x}

2 3^{n} = - (5^{n})

7^x-7=35+7^{3-x}

7^{x} - 7 = 35 + 7^{3 - x}

3^{5 x - 3} = 21