设logaC,logbC是方程x^2-3x+1=0的两根,求log(a/b)C的值

Question

设logaC,logbC是方程x^2-3x+1=0的两根,求log(a/b) C的值

Answer 1

即loga(c)+logb(c)=3
loga(c)logb(c)=1
lgc/lga+lgc/lgb=3
lg²c/lgalgb=1
lg²c=lgalgb
lgc/lga+lgc/lgb=3
lgc(lga+lgb)/lgalgb=3
lgc(lga+lgb)/lg²c=3
lga+lgb=3lgc
平方
lg²a+2lgalgb+lg²b=9lg²c
两边减4lgalgb=lg²c
lg²a-2lgalgb+lg²b=5lg²c
(lga-lgb)²=5lg²c
lga-lgb=±√5lgc
lg(a/b)=±√5lgc
原式用
=lgc/lg(a/b)=±√5/5