A Question from Series and Sequence.

let first term of G.P. be A and geometric ratio be R

so α = AR^p-1

β = AR^q-1

γ = AR^r-1

α^q-r · β^r-p · γ^p-q = A^{(q-r+r-p+p-q)} R^{(pq-pr-r+q + qr-pq -r+p+pr-qr - p+q)} = A⁰R⁰ = 1

now taking log on both sides

(q-r) log α + (r-p) log β + (p-q) log γ = 0.

here we have used the formulas log(a.b.c) = loga + logb +logc

and log aⁿ = n loga