Problem: Let a,b and x be positive real numbers distinct from one. Then
4(loga​x)2+3(logb​x)2=8(loga​x)(logb​x)
Answer Choices:
A. For all values of a,b and x
B. If and only if a=b2
C. If and only if b=a2
D. If and only if x=ab
E. For none of these
Solution:
The given equation may be written in the form
4(loga​x)2−8(loga​x)(logb​x)+3(logb​x)2=0(2loga​x−logb​x)(2loga​x−3logb​x)=0loga​x2=logb​x or loga​x2=logb​x3​
Let r=loga​x2. Then
ar=x2 and br=x, or ar=x2 and br=x3;ar=b2r or a3r=b2r;a=b2 or a3=b2.​