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受欢迎的代数 >

x^3=2^x

解答

x^{3} = 2^{x}

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

x^{3} = 2^{x}

将

x^{3} = 2^{x}

调整为朗伯形式:

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

用

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

和

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

改写方程式

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

改写

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

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

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3^{x + 5} + 3^{x} = 177876

a^{x} = x

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