若以10为底A的对数和以10为底B的对数是2x^2-4x+1=0的两个根,求以10为底（A*B）乘以以10为底（A/B）的值

Question

若以10为底A的对数和以10为底B的对数是2x^2-4x+1=0的两个根,求以10为底（A*B）乘以以10为底（A/B）的值

Answer 1

由2x^2 -- 4x + 1=0 ,解得x1 = 1 + (根号2)/2,x2 = 1 - (根号2)/2
令x1 = lgA = 1 + (根号2)/2,x2 = lgB = 1 - (根号2)/2
lg(AB) ．lg(A/B)
= (lg A + lg B) (lg A - lg B)
=(x1 + x2) (x1 - x2)
=[1 + (根号2)/2 + 1 - (根号2)/2] {1 + (根号2)/2 –[ 1 - (根号2)/2]}
= 2(根号2)