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受欢迎的代数 >

8^3x-1=7^x+2

解答

8^{3} x - 1 = 7^{x} + 2

解答

x = - \frac{512 W _{- 1} ( - \frac{e ^{\frac{3 l n ( 7 )}{512}} l n ( 7 )}{512} ) - 3 ln ( 7 )}{512 ln ( 7 )}, x = - \frac{512 W _{0} ( - \frac{e ^{\frac{3 l n ( 7 )}{512}} l n ( 7 )}{512} ) - 3 ln ( 7 )}{512 ln ( 7 )}

+1

十进制

x = 3.90517 \dots, x = 0.00784 \dots

求解步骤

8^{3} x - 1 = 7^{x} + 2

将

8^{3} x - 1 = 7^{x} + 2

调整为朗伯形式:

1 = (512 x - 3) e^{- l n (7) x}

用

(- x + \frac{3}{512}) ln (7) = u

和

x = - \frac{512 u - 3 ln ( 7 )}{512 ln ( 7 )}

改写方程式

1 = (512 (- \frac{512 u - 3 ln ( 7 )}{512 ln ( 7 )}) - 3) e^{- l n (7) (- \frac{512 u - 3 l n ( 7 )}{512 l n ( 7 )})}

化简

1 = e^{\frac{512 u - 3 l n ( 7 )}{512}} (- \frac{512 u - 3 ln ( 7 )}{ln ( 7 )} - 3)

改写

1 = e^{\frac{512 u - 3 l n ( 7 )}{512}} (- \frac{512 u - 3 ln ( 7 )}{ln ( 7 )} - 3)

为朗伯形式:

e^{u} u = - \frac{e ^{\frac{3 l n ( 7 )}{512}} ln ( 7 )}{512}

解

e^{u} u = - \frac{e ^{\frac{3 l n ( 7 )}{512}} ln ( 7 )}{512} : u = W_{- 1} (- \frac{e ^{\frac{3 l n ( 7 )}{512}} ln ( 7 )}{512}), u = W_{0} (- \frac{e ^{\frac{3 l n ( 7 )}{512}} ln ( 7 )}{512})

u = W_{- 1} (- \frac{e ^{\frac{3 l n ( 7 )}{512}} ln ( 7 )}{512}), u = W_{0} (- \frac{e ^{\frac{3 l n ( 7 )}{512}} ln ( 7 )}{512})

代回

u = (- x + \frac{3}{512}) ln (7)

，求解

x

解

(- x + \frac{3}{512}) ln (7) = W_{- 1} (- \frac{e ^{\frac{3 l n ( 7 )}{512}} ln ( 7 )}{512}) : x = - \frac{512 W _{- 1} ( - \frac{e ^{\frac{3 l n ( 7 )}{512}} l n ( 7 )}{512} ) - 3 ln ( 7 )}{512 ln ( 7 )}

解

(- x + \frac{3}{512}) ln (7) = W_{0} (- \frac{e ^{\frac{3 l n ( 7 )}{512}} ln ( 7 )}{512}) : x = - \frac{512 W _{0} ( - \frac{e ^{\frac{3 l n ( 7 )}{512}} l n ( 7 )}{512} ) - 3 ln ( 7 )}{512 ln ( 7 )}

x = - \frac{512 W _{- 1} ( - \frac{e ^{\frac{3 l n ( 7 )}{512}} l n ( 7 )}{512} ) - 3 ln ( 7 )}{512 ln ( 7 )}, x = - \frac{512 W _{0} ( - \frac{e ^{\frac{3 l n ( 7 )}{512}} l n ( 7 )}{512} ) - 3 ln ( 7 )}{512 ln ( 7 )}

作图

流行的例子

2^{2x+1}-33(2^{x-1})+4=0

2^{2 x + 1} - 33 (2^{x - 1}) + 4 = 0

5^{x-3}+125^{x+3}=78125

5^{x - 3} + 12 5^{x + 3} = 78125

(\frac{1}{2})^{n} + 1 = 2

solvefor t,4((x^2)^{-t})/5 =288

so l v e f or t, 4 \frac{( x ^{2} ) ^{- t}}{5} = 288

2^{x} + 3 \cdot 2^{x} = 16