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受欢迎的代数 >

5^{x+1}+5x=6

解答

5^{x + 1} + 5 x = 6

解答

x = - \frac{5 W _{0} ( \frac{e ^{\frac{11 l n ( 5 )}{5}} l n ( 5 )}{5} ) - 6 ln ( 5 )}{5 ln ( 5 )}

+1

十进制

x = 0.07382 \dots

求解步骤

5^{x + 1} + 5 x = 6

将

5^{x + 1} + 5 x = 6

调整为朗伯形式:

1 = (6 - 5 x) e^{l n (5) (- x - 1)}

用

(- x + \frac{6}{5}) ln (5) = u

和

x = - \frac{5 u - 6 ln ( 5 )}{5 ln ( 5 )}

改写方程式

1 = (6 - 5 (- \frac{5 u - 6 ln ( 5 )}{5 ln ( 5 )})) e^{l n (5) (- (- \frac{5 u - 6 l n ( 5 )}{5 l n ( 5 )}) - 1)}

化简

1 = e^{\frac{5 u - 6 l n ( 5 )}{5} - l n (5)} (6 + \frac{5 u - 6 ln ( 5 )}{ln ( 5 )})

改写

1 = e^{\frac{5 u - 6 l n ( 5 )}{5} - l n (5)} (6 + \frac{5 u - 6 ln ( 5 )}{ln ( 5 )})

为朗伯形式:

e^{u} u = \frac{e ^{\frac{11 l n ( 5 )}{5}} ln ( 5 )}{5}

解

e^{u} u = \frac{e ^{\frac{11 l n ( 5 )}{5}} ln ( 5 )}{5} : u = W_{0} (\frac{e ^{\frac{11 l n ( 5 )}{5}} ln ( 5 )}{5})

u = W_{0} (\frac{e ^{\frac{11 l n ( 5 )}{5}} ln ( 5 )}{5})

代回

u = (- x + \frac{6}{5}) ln (5)

，求解

x

解

(- x + \frac{6}{5}) ln (5) = W_{0} (\frac{e ^{\frac{11 l n ( 5 )}{5}} ln ( 5 )}{5}) : x = - \frac{5 W _{0} ( \frac{e ^{\frac{11 l n ( 5 )}{5}} l n ( 5 )}{5} ) - 6 ln ( 5 )}{5 ln ( 5 )}

x = - \frac{5 W _{0} ( \frac{e ^{\frac{11 l n ( 5 )}{5}} l n ( 5 )}{5} ) - 6 ln ( 5 )}{5 ln ( 5 )}

作图

流行的例子

4(3^x)=3(6^{(3x-3)})

4 (3^{x}) = 3 (6^{(3 x - 3)})

71 = 2 7^{x}

4^{- 3 t} = 0.1

4^{(2x+1)}=1024

4^{(2 x + 1)} = 1024

3^{x} = e^{x + 2}