Given that a,b,c are p‌th ‌,q‌th ‌ and r‌th ‌ terms of a G.P. then,
Ap=ARp−1=a,Aq=ARq−1=b,Ar=ARr−1=c,
where A and R are the 1‌st ‌ term and common ratio of the geometric progression respectively.
Consider LHS : Let ∆=|

log‌a	p	1
log‌b	q	1
log‌c	r	1

|
⇒∆=|

log[ARp−1]	p	1
log[ARq−1]	q	1
log[ARr−1]	r	1

| [

∵log(mn)=log‌m+log‌n

log(m)n=n‌log‌m

.
∴∆=|

log‌A+(p−1)‌log‌R	p	1
log‌A+(q−1)‌log‌R	q	1
log‌A+(r−1)‌log‌R	r	1

|
By C1→C1−(log‌A)C3
⇒∆=|

(p−1)‌log‌R	p	1
(q−1)‌log‌R	q	1
(r−1)‌log‌R	r	1

|
Taking log‌R common from C1
⇒∆=log‌R‌|

p−1	p	1
q−1	q	1
r−1	r	1

|
By C1→C1+C3
⇒∆=log‌R‌|

p	p	1
q	q	1
r	r	1

|
Since C1 and C2 are identical, ∴∆=0= RHS.