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