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受欢迎的代数 >

3^{2x-3}=27(x+1)/3

解答

3^{2 x - 3} = 27 \frac{x + 1}{3}

解答

x = - \frac{W _{- 1} ( - \frac{2 l n ( 3 )}{2187} ) + 2 ln ( 3 )}{2 ln ( 3 )}, x = - \frac{W _{0} ( - \frac{2 l n ( 3 )}{2187} ) + 2 ln ( 3 )}{2 ln ( 3 )}

+1

十进制

x = 3.14739 \dots, x = - 0.99954 \dots

求解步骤

3^{2 x - 3} = 27 \cdot \frac{x + 1}{3}

将

3^{2 x - 3} = 27 \frac{x + 1}{3}

调整为朗伯形式:

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

用

(- 2 x - 2) ln (3) = u

和

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

改写方程式

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

化简

1 = e^{u + 7 l n (3)} (- \frac{u + 2 ln ( 3 )}{2 ln ( 3 )} + 1)

改写

1 = e^{u + 7 l n (3)} (- \frac{u + 2 ln ( 3 )}{2 ln ( 3 )} + 1)

为朗伯形式:

e^{u} u = - \frac{2 ln ( 3 )}{2187}

解

e^{u} u = - \frac{2 ln ( 3 )}{2187} : u = W_{- 1} (- \frac{2 ln ( 3 )}{2187}), u = W_{0} (- \frac{2 ln ( 3 )}{2187})

u = W_{- 1} (- \frac{2 ln ( 3 )}{2187}), u = W_{0} (- \frac{2 ln ( 3 )}{2187})

代回

u = (- 2 x - 2) ln (3)

，求解

x

解

(- 2 x - 2) ln (3) = W_{- 1} (- \frac{2 ln ( 3 )}{2187}) : x = - \frac{W _{- 1} ( - \frac{2 l n ( 3 )}{2187} ) + 2 ln ( 3 )}{2 ln ( 3 )}

解

(- 2 x - 2) ln (3) = W_{0} (- \frac{2 ln ( 3 )}{2187}) : x = - \frac{W _{0} ( - \frac{2 l n ( 3 )}{2187} ) + 2 ln ( 3 )}{2 ln ( 3 )}

x = - \frac{W _{- 1} ( - \frac{2 l n ( 3 )}{2187} ) + 2 ln ( 3 )}{2 ln ( 3 )}, x = - \frac{W _{0} ( - \frac{2 l n ( 3 )}{2187} ) + 2 ln ( 3 )}{2 ln ( 3 )}

作图

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