The terms of a geometric progression are real and positive. If the p-th term of the progression is q and the q-th term is p, then the logarithm of the first term is

Given a series in GP which are positive real numbers ..Let the first term be "a" and common ratio be "r".
$$p^{th}$$ term =$$a\times r^{p-1}$$ = q 
$$q^{th}$$ term =$$a\times r^{q-1}$$ = p
By dividing both, we get $$r=\ \left(\dfrac{\ q}{p}\right)^{\ \dfrac{\ 1}{p-q}}$$
By substituting this in 2nd equation , we get : $$a\left(\ \dfrac{\ q}{p}\right)^{\ \dfrac{q-1\ }{p-q}}=p$$
Now applying log on both sides :
Log a + $$\dfrac{\ q-1}{p-1}\left(\log\dfrac{\ q}{p}\right)$$ = log p
Therefore , log a = log p $$-$$ $$\ \dfrac{\ q-1}{p-q}\left(\log\ q-\log\ p\right)$$

Hence, log a = $$\ \dfrac{\ \left(p-1\right)\log\ p\ +\ \left(q-1\right)\log\ q}{p-q}$$