设loga(c),logb(c)是方程x^2-3x+1=0的两根,求loga/b(c)的值

设loga(c),logb(c)是方程x^2-3x+1=0的两根,求loga/b(c)的值

loga(c),logb(c)是方程x^2-3x+1=0的两根
loga(c)+logb(c)=3
loga(c)*logb(c)=1
logc(a)+logc(b)=1/loga(c)+1/logb(c)=[loga(c)+logb(c)]/loga(c)*logb(c)=3
logc(a)*logc(b)=1/loga(c)*1/logb(c)=1
(logc(a)-logc(b))^2
=(logc(a)+logc(b))^2-4logc(a)*logc(b)
=3^2-4*1
=9-4
=5
logc(a)-logc(b)=±√5
loga/b(c)
=lgc/lg(a/b)
=lgc/(lga-lgb)
=1/(lga/lgc-lgb/lgc)
=1/(logc(a)-logc(b))
=1/±√5
=±√5/5