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受欢迎的代数 >

2^2x-2^x+3+2^4=0

解答

2^{2} x - 2^{x} + 3 + 2^{4} = 0

解答

x = - \frac{4 W _{- 1} ( - \frac{l n ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}} ) + 19 ln ( 2 )}{4 ln ( 2 )}, x = - \frac{4 W _{0} ( - \frac{l n ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}} ) + 19 ln ( 2 )}{4 ln ( 2 )}

+1

十进制

x = 5.33399 \dots, x = - 4.74064 \dots

求解步骤

2^{2} x - 2^{x} + 3 + 2^{4} = 0

将

2^{2} x - 2^{x} + 3 + 2^{4} = 0

调整为朗伯形式:

- 1 = - e^{- l n (2) x} (4 x + 19)

用

(- x - \frac{19}{4}) ln (2) = u

和

x = - \frac{4 u + 19 ln ( 2 )}{4 ln ( 2 )}

改写方程式

- 1 = - e^{- l n (2) (- \frac{4 u + 19 l n ( 2 )}{4 l n ( 2 )})} (4 (- \frac{4 u + 19 ln ( 2 )}{4 ln ( 2 )}) + 19)

化简

- 1 = - e^{\frac{4 u + 19 l n ( 2 )}{4}} (- \frac{4 u + 19 ln ( 2 )}{ln ( 2 )} + 19)

改写

- 1 = - e^{\frac{4 u + 19 l n ( 2 )}{4}} (- \frac{4 u + 19 ln ( 2 )}{ln ( 2 )} + 19)

为朗伯形式:

\frac{e ^{\frac{4 u + 19 l n ( 2 )}{4}} u}{e ^{\frac{19 l n ( 2 )}{4}}} = - \frac{ln ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}}

解

\frac{e ^{\frac{4 u + 19 l n ( 2 )}{4}} u}{e ^{\frac{19 l n ( 2 )}{4}}} = - \frac{ln ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}} : u = W_{- 1} (- \frac{ln ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}}), u = W_{0} (- \frac{ln ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}})

u = W_{- 1} (- \frac{ln ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}}), u = W_{0} (- \frac{ln ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}})

代回

u = (- x - \frac{19}{4}) ln (2)

，求解

x

解

(- x - \frac{19}{4}) ln (2) = W_{- 1} (- \frac{ln ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}}) : x = - \frac{4 W _{- 1} ( - \frac{l n ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}} ) + 19 ln ( 2 )}{4 ln ( 2 )}

解

(- x - \frac{19}{4}) ln (2) = W_{0} (- \frac{ln ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}}) : x = - \frac{4 W _{0} ( - \frac{l n ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}} ) + 19 ln ( 2 )}{4 ln ( 2 )}

x = - \frac{4 W _{- 1} ( - \frac{l n ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}} ) + 19 ln ( 2 )}{4 ln ( 2 )}, x = - \frac{4 W _{0} ( - \frac{l n ( 2 )}{4 e ^{\frac{19 l n ( 2 )}{4}}} ) + 19 ln ( 2 )}{4 ln ( 2 )}

作图

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