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受欢迎的代数 >

3^{x-3}=2x-5

解答

3^{x - 3} = 2 x - 5

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 - 3} = 2 x - 5

将

3^{x - 3} = 2 x - 5

调整为朗伯形式:

1 = (2 x - 5) e^{l n (3) (- x + 3)}

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

化简

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

改写

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

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

流行的例子

4x^3*e^{-3x}-3x^4*e^{-3x}=0

4 x^{3} \cdot e^{- 3 x} - 3 x^{4} \cdot e^{- 3 x} = 0

(2^{x+5})/(3^{x+5)}=0.01156

\frac{2 ^{x + 5}}{3 ^{x + 5}} = 0.01156

3^{x - 1} = 27

3^{2 x - 6} = 9^{4}

x^{log_{5}(x)}=9

x^{l o g_{5} (x)} = 9