很急!!高中數學求解log的運算!

👨🏻‍🏫 學術台數學

安那蘇姆

2014-01-03 3:28 am

設log x+ log y = 2,且x^2+y^2=425則x+y=?
10 10

註明: log =>以10為底
10

X^2 =>x的2次方

更新1:

設log x+ log y = 2,且x^2+y^2=425則x+y=? 10 10

更新2:

設log x+ log y = 2,且x^2+y^2=425則x+y=? ....... 10 ....10......

回答 (2)

螞蟻雄兵

2014-01-03 3:53 am

✔ 最佳答案

設log x+ log y= 2,且x^2+y^2=425則x+y=?
Sol
習慣上以10為底不必寫
log x+ log y = 2
log(xy)=log100
xy=100
y=100/x
425=x^2+y^2
=x^2+(100/x)^2
=x^2+10000/x^2
425x^2=x^4+10000
x^4－425x^2+10000=0
(x^2－25)(x^2－400)=0
x^2=25 or x^2=400
x=5 or x=－5(不合) or x=20 or x=－20(不合)
(1) x=5
y=20
x+y=25
(2) x=20
y=5
x+y=25
So
x+y=25

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Lopez

2014-01-03 3:58 am

令 a = log x , b = log y

故 a+b = 2
x = 10^a
y = 10^b = 10^(2-a) = 10^2 / 10^a = 100 / x

故 xy = x (100 / x) = 100

(x+y)^2
= x^2 + y^2 + 2xy
= 425 +2*100
= 625
= 25^2

x+y =±25 (負不合, 因為x+y = 10^a+10^b > 0 )

Ans: 25

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