Posted by Anonymous on .
of a^2+b^2=2ab, prove log[(a+b)/2]=1/2(loga+logb)

math 
kayla,
systems of equations by graphing

math 
MathMate,
I have made some editorial changes to reflect my interpretation of the question:
"If a^2+b^2=2ab,
prove log[(a+b)/2]=(1/2)(loga+logb)"
If a²+b²=2ab, then
a²+b²2ab=0
(ab)²=0 after factoring
So we conclude that a=b
Substituting a=b into the lefthand side of the equation:
log[(a+b)/2]
=log((a+a)/2)
=log(a)
The righthand side:
(1/2)(loga+logb)
=(1/2)(log(a)+log(a))
=log(a)
Therefore:
log[(a+b)/2]=(1/2)(loga+logb)