Question: If and
, pyrove that:
Solution: This is same as proving: y is Harmonic Mean (HM) of x and z;
That is, to prove that is the same as the proof for :
Now, it is given that —– I
and —– II
Let say. By definition of logarithm,
;
;
;
;
.
Now let us see what happens to the following two algebraic entities, namely, and
;
Now, …call this III
Now,
Hence, ….equation IV
but it is also given that …see equation II
Hence,
Take log of above both sides w.r.t. base N:
So, above is equivalent to
But now see relations III and IV:
Hence,
Hence,
Hence, as desired.
Regards,
Nalin Pithwa