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受欢迎的代数 >

2^{x-10}=x^3

解答

2^{x - 10} = x^{3}

解答

x = - \frac{3 W _{- 1} ( - \frac{l n ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}} )}{ln ( 2 )}, x = - \frac{3 W _{0} ( - \frac{l n ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}} )}{ln ( 2 )}

+1

十进制

x = 23.70054 \dots, x = 0.10156 \dots

求解步骤

2^{x - 10} = x^{3}

将

2^{x - 10} = x^{3}

调整为朗伯形式:

1 = x e^{- \frac{l n ( 2 ) ( x - 10 )}{3}}

用

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

和

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

改写方程式

1 = (- \frac{3 u}{ln ( 2 )}) e^{- \frac{l n ( 2 ) ( ( - \frac{3 u}{l n ( 2 )} ) - 10 )}{3}}

化简

1 = - \frac{3 e ^{- \frac{- 3 u - 10 l n ( 2 )}{3}} u}{ln ( 2 )}

改写

1 = - \frac{3 e ^{- \frac{- 3 u - 10 l n ( 2 )}{3}} u}{ln ( 2 )}

为朗伯形式:

\frac{e ^{- \frac{- 3 u - 10 l n ( 2 )}{3}} u}{e ^{\frac{10 l n ( 2 )}{3}}} = - \frac{ln ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}}

解

\frac{e ^{- \frac{- 3 u - 10 l n ( 2 )}{3}} u}{e ^{\frac{10 l n ( 2 )}{3}}} = - \frac{ln ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}} : u = W_{- 1} (- \frac{ln ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}}), u = W_{0} (- \frac{ln ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}})

u = W_{- 1} (- \frac{ln ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}}), u = W_{0} (- \frac{ln ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}})

代回

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

，求解

x

解

- \frac{1}{3} x ln (2) = W_{- 1} (- \frac{ln ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}}) : x = - \frac{3 W _{- 1} ( - \frac{l n ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}} )}{ln ( 2 )}

解

- \frac{1}{3} x ln (2) = W_{0} (- \frac{ln ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}}) : x = - \frac{3 W _{0} ( - \frac{l n ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}} )}{ln ( 2 )}

x = - \frac{3 W _{- 1} ( - \frac{l n ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}} )}{ln ( 2 )}, x = - \frac{3 W _{0} ( - \frac{l n ( 2 )}{3 e ^{\frac{10 l n ( 2 )}{3}}} )}{ln ( 2 )}

作图

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