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受欢迎的代数 >

2^x-1=x+2

解答

2^{x} - 1 = x + 2

解答

x = - \frac{W _{- 1} ( - \frac{l n ( 2 )}{8} ) + 3 ln ( 2 )}{ln ( 2 )}, x = - \frac{W _{0} ( - \frac{l n ( 2 )}{8} ) + 3 ln ( 2 )}{ln ( 2 )}

+1

十进制

x = 2.44490 \dots, x = - 2.86250 \dots

求解步骤

2^{x} - 1 = x + 2

将

2^{x} - 1 = x + 2

调整为朗伯形式:

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

用

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

和

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

改写方程式

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

化简

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

改写

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

为朗伯形式:

e^{u} u = - \frac{ln ( 2 )}{8}

解

e^{u} u = - \frac{ln ( 2 )}{8} : u = W_{- 1} (- \frac{ln ( 2 )}{8}), u = W_{0} (- \frac{ln ( 2 )}{8})

u = W_{- 1} (- \frac{ln ( 2 )}{8}), u = W_{0} (- \frac{ln ( 2 )}{8})

代回

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

，求解

x

解

(- x - 3) ln (2) = W_{- 1} (- \frac{ln ( 2 )}{8}) : x = - \frac{W _{- 1} ( - \frac{l n ( 2 )}{8} ) + 3 ln ( 2 )}{ln ( 2 )}

解

(- x - 3) ln (2) = W_{0} (- \frac{ln ( 2 )}{8}) : x = - \frac{W _{0} ( - \frac{l n ( 2 )}{8} ) + 3 ln ( 2 )}{ln ( 2 )}

x = - \frac{W _{- 1} ( - \frac{l n ( 2 )}{8} ) + 3 ln ( 2 )}{ln ( 2 )}, x = - \frac{W _{0} ( - \frac{l n ( 2 )}{8} ) + 3 ln ( 2 )}{ln ( 2 )}

作图

流行的例子

x e^{2 x} = 0

2^{2x+1}*4^{3x+1}=8^{x-1}

2^{2 x + 1} \cdot 4^{3 x + 1} = 8^{x - 1}

e^{x} + e^{- x} = 0

e^{e^{x}} = 10

2^{3 x} = 256