对数方程如果方程lg^2x-lgx^2-2=0有两根a,b,那么求logab+logba的值(写的详细点)

Question

对数方程
如果方程lg^2 x - lgx^2 -2=0有两根a,b,那么求loga b + logb a的值(写的详细点)

Answer 1

(lgx)^2 - 2lgx -2=0
===>(lgx-1)^2-3=0
===>(lgx-1+√3)(lgx-1-√3)=0
===>lga=1-√3,lgb=1+√3
loga b + logb a
=lgb/lga+lga/lgb
=((lgb)^2+(lga)^2)/(lgalgb)
=[(1-√3)^2+(1+√3)^2]/[(1+√3)(1-√3)]
=-4