數學ln計算

125/225=(1－e^－2k)/(1－e^－8k)
Sol
Set x=e^(－2k)
x^4=e^(－8k)
125/225=(1－x)/(1－x^4)
5/9=(1－x)/(1－x^4)
9－9x=5－5x^4
5x^4－9x+4=0
(5x^4－5x^3)+(5x^3－5x^2)+(5x^2－5x)－(4x－4)=0
(x－1)(5x^3+5x^2+5x－4)=0
5x^3+5x^2+5x－4
=(5x^3－2.36333x^2)+(7.36333x^2－3.4803957x)+(8.4803957x－4)
=(5x^2+7.36333x+8.4803957)(x－0.472666)
D=7.36333^2－4*5*8.4803957=－115.38929
So
5x^4－9x+4=0
(x－1)(x－0.472666)(5x^2+7.36333x+8.4803957)=0
有2實根1和0.472666
(1) x=1 (分母為0不合)
(2) x=0.472666
0.472666=e^(－2k)
－2k=ln(0.472666)
－2k=－0.749366
k=0.374683

Let x = exp(-2k)

125/225 = (1 - x)/(1 - x^4) = 1/(1 + x)(1 + x^2)

5/9 = 1/(1 + x)(1 + x^2)

5(1 + x)(1 + x^2) = 9

5(1 + x^2 + x + x^3) = 9

5x^3 + 5x^2 + 5x - 4 = 0

x = 0.47202

exp(-2k) = 0.47202

k = 0.3754

....恕我看不懂你的題目@@
(拍照直接傳應該會好點 = =)
但 還記得高中的log嗎?
ln(X)=以e為底的log e (x)
但願能幫到你@@