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受欢迎的代数 >

3^x-54x+135=0

解答

3^{x} - 54 x + 135 = 0

解答

x = - \frac{2 W _{- 1} ( - \frac{e ^{\frac{5 l n ( 3 )}{2}} l n ( 3 )}{54} ) - 5 ln ( 3 )}{2 ln ( 3 )}, x = - \frac{2 W _{0} ( - \frac{e ^{\frac{5 l n ( 3 )}{2}} l n ( 3 )}{54} ) - 5 ln ( 3 )}{2 ln ( 3 )}

+1

十进制

x = 4, x = 3

求解步骤

3^{x} - 54 x + 135 = 0

将

3^{x} - 54 x + 135 = 0

调整为朗伯形式:

1 = - e^{- l n (3) x} (- 54 x + 135)

用

(- x + \frac{5}{2}) ln (3) = u

和

x = - \frac{2 u - 5 ln ( 3 )}{2 ln ( 3 )}

改写方程式

1 = - e^{- l n (3) (- \frac{2 u - 5 l n ( 3 )}{2 l n ( 3 )})} (- 54 (- \frac{2 u - 5 ln ( 3 )}{2 ln ( 3 )}) + 135)

化简

1 = - e^{\frac{2 u - 5 l n ( 3 )}{2}} (\frac{27 ( 2 u - 5 ln ( 3 ) )}{ln ( 3 )} + 135)

改写

1 = - e^{\frac{2 u - 5 l n ( 3 )}{2}} (\frac{27 ( 2 u - 5 ln ( 3 ) )}{ln ( 3 )} + 135)

为朗伯形式:

e^{u} u = - \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54}

解

e^{u} u = - \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54} : u = W_{- 1} (- \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54}), u = W_{0} (- \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54})

u = W_{- 1} (- \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54}), u = W_{0} (- \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54})

代回

u = (- x + \frac{5}{2}) ln (3)

，求解

x

解

(- x + \frac{5}{2}) ln (3) = W_{- 1} (- \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54}) : x = - \frac{2 W _{- 1} ( - \frac{e ^{\frac{5 l n ( 3 )}{2}} l n ( 3 )}{54} ) - 5 ln ( 3 )}{2 ln ( 3 )}

解

(- x + \frac{5}{2}) ln (3) = W_{0} (- \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54}) : x = - \frac{2 W _{0} ( - \frac{e ^{\frac{5 l n ( 3 )}{2}} l n ( 3 )}{54} ) - 5 ln ( 3 )}{2 ln ( 3 )}

x = - \frac{2 W _{- 1} ( - \frac{e ^{\frac{5 l n ( 3 )}{2}} l n ( 3 )}{54} ) - 5 ln ( 3 )}{2 ln ( 3 )}, x = - \frac{2 W _{0} ( - \frac{e ^{\frac{5 l n ( 3 )}{2}} l n ( 3 )}{54} ) - 5 ln ( 3 )}{2 ln ( 3 )}

作图

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