［數學］關於指對數之根與係數關係

Question

<1> 設 a、b 為方程式 (log2x)(log3x) = 1 的兩根，則 ab =? Ans: 1 / 6

<2> 設 C= 2^X，且 m、n 為 C^2 - 10 C + 8 = 0的兩根，則m+n=? Ans: 3

請問該如何運用根與係數關係求解呢？！

Answer 1

1. (log2x)(log3x) = 1
(log2 + logx)(log3 + logx) = 1
(logx)2 + (log2+log3)logx +log2*log3-1= 0
令 A= logx => A2 + (log6)A +log2*log3-1= 0, 兩根為 log(a), log(b)
=> log(a)+ log(b) = - log 6
=> log(ab) = log(1/6)
=> ab = 1/6
2. C= 2x => C2 - 10 C + 8 = 0 , 兩根為 2m, 2n
=> 2m * 2n = 8
=> 2m+n = 23
=> m+n = 3
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