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受欢迎的代数 >

f^5=5^f

解答

f^{5} = 5^{f}

解答

f = 5, f = - \frac{5 W _{0} ( - \frac{l n ( 5 )}{5} )}{ln ( 5 )}

+1

十进制

f = 5, f = 1.76492 \dots

求解步骤

f^{5} = 5^{f}

将

f^{5} = 5^{f}

调整为朗伯形式:

f e^{- \frac{l n ( 5 ) f}{5}} = 1

用

- \frac{ln ( 5 ) f}{5} = u

和

f = - \frac{5 u}{ln ( 5 )}

改写方程式

(- \frac{5 u}{ln ( 5 )}) e^{u} = 1

改写

(- \frac{5 u}{ln ( 5 )}) e^{u} = 1

为朗伯形式:

e^{u} u = - \frac{ln ( 5 )}{5}

解

e^{u} u = - \frac{ln ( 5 )}{5} : u = - ln (5), u = W_{0} (- \frac{ln ( 5 )}{5})

u = - ln (5), u = W_{0} (- \frac{ln ( 5 )}{5})

代回

u = - \frac{ln ( 5 ) f}{5}

，求解

f

解

- \frac{ln ( 5 ) f}{5} = - ln (5) : f = 5

解

- \frac{ln ( 5 ) f}{5} = W_{0} (- \frac{ln ( 5 )}{5}) : f = - \frac{5 W _{0} ( - \frac{l n ( 5 )}{5} )}{ln ( 5 )}

f = 5, f = - \frac{5 W _{0} ( - \frac{l n ( 5 )}{5} )}{ln ( 5 )}

作图

流行的例子

2(3.1)^x=(6.6)^x

2 (3.1)^{x} = (6.6)^{x}

m^{x + 1} = \frac{1}{2}

25^{4x+1}=5^{x-5}

2 5^{4 x + 1} = 5^{x - 5}

2 \cdot e^{- 6 x} = 95

5^{x^2}*25^{x-1}=5^{3x}

5^{x^{2}} \cdot 2 5^{x - 1} = 5^{3 x}