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受欢迎的代数 >

e^x=sqrt(x+2)

解答

e^{x} = x + 2

解答

x = - \frac{W _{- 1} ( - \frac{2}{e ^{4}} ) + 4}{2}, x = - \frac{W _{0} ( - \frac{2}{e ^{4}} ) + 4}{2}

+1

十进制

x = 0.44754 \dots, x = - 1.98097 \dots

求解步骤

e^{x} = x + 2

将

e^{x} = x + 2

调整为朗伯形式:

1 = (x + 2) e^{- 2 x}

用

- 2 x - 4 = u

和

x = - \frac{u + 4}{2}

改写方程式

1 = ((- \frac{u + 4}{2}) + 2) e^{- 2 (- \frac{u + 4}{2})}

化简

1 = e^{u + 4} (- \frac{u + 4}{2} + 2)

改写

1 = e^{u + 4} (- \frac{u + 4}{2} + 2)

为朗伯形式:

e^{u} u = - \frac{2}{e ^{4}}

解

e^{u} u = - \frac{2}{e ^{4}} : u = W_{- 1} (- \frac{2}{e ^{4}}), u = W_{0} (- \frac{2}{e ^{4}})

u = W_{- 1} (- \frac{2}{e ^{4}}), u = W_{0} (- \frac{2}{e ^{4}})

代回

u = - 2 x - 4

，求解

x

解

- 2 x - 4 = W_{- 1} (- \frac{2}{e ^{4}}) : x = - \frac{W _{- 1} ( - \frac{2}{e ^{4}} ) + 4}{2}

解

- 2 x - 4 = W_{0} (- \frac{2}{e ^{4}}) : x = - \frac{W _{0} ( - \frac{2}{e ^{4}} ) + 4}{2}

x = - \frac{W _{- 1} ( - \frac{2}{e ^{4}} ) + 4}{2}, x = - \frac{W _{0} ( - \frac{2}{e ^{4}} ) + 4}{2}

验证解:

x = - \frac{W _{- 1} ( - \frac{2}{e ^{4}} ) + 4}{2}

真

, x = - \frac{W _{0} ( - \frac{2}{e ^{4}} ) + 4}{2}

真

解为

x = - \frac{W _{- 1} ( - \frac{2}{e ^{4}} ) + 4}{2}, x = - \frac{W _{0} ( - \frac{2}{e ^{4}} ) + 4}{2}

验证解:

x = - \frac{W _{- 1} ( - \frac{2}{e ^{4}} ) + 4}{2}

真

, x = - \frac{W _{0} ( - \frac{2}{e ^{4}} ) + 4}{2}

真

解为

x = - \frac{W _{- 1} ( - \frac{2}{e ^{4}} ) + 4}{2}, x = - \frac{W _{0} ( - \frac{2}{e ^{4}} ) + 4}{2}

作图

流行的例子

e^{x} = - 7

2^{2 x - 1} = \frac{1}{8}

e^{7x+7}=7e^{7-x}

e^{7 x + 7} = 7 e^{7 - x}

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

(\frac{2}{3})^{x} = 2^{x - 1}

27(3^{2x})+80(3^x)-3=0

27 (3^{2 x}) + 80 (3^{x}) - 3 = 0