Problem: Let a,b and x be positive real numbers distinct from one. Then
4(logax)2+3(logbx)2=8(logax)(logbx)
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(logax)2−8(logax)(logbx)+3(logbx)2=0(2logax−logbx)(2logax−3logbx)=0logax2=logbx or logax2=logbx3
Let r=logax2. Then
ar=x2 and br=x, or ar=x2 and br=x3;ar=b2r or a3r=b2r;a=b2 or a3=b2.