x, y, z are three integers in a geometric progression (G.P.) such that y − x…

Concept: two identities unlock a nested logarithm equation like this one. First, the power-base rule: logaⁿ(N) = (1/n) loga(N).

Second, for three integers a, ar, ar² in a G.P., their product is the cube of the middle term: a · (ar) · (ar²) = (ar)³, so the product of any three G.P. terms always equals (middle term)³.

Applying these to the equation, step by step:

1. Convert every log term to base 6 using the power-base rule: log36(x2) = log62(x2) = (2/2) log6(x) = log6(x).

2. log6(√y) = (1/2) log6(y).

3. 3 log216(√y · z) = 3 log63(√y · z) = 3 · (1/3) log6(√y · z) = log6(√y · z) = (1/2) log6(y) + log6(z).

4. Add the three rewritten terms: log6(x) + (1/2) log6(y) + (1/2) log6(y) + log6(z) = log6(x) + log6(y) + log6(z) = log6(xyz).

5. Set this equal to the right-hand side: log6(xyz) = 6, so xyz = 66.

6. Since x, y, z are in G.P., xyz equals the cube of the middle term: xyz = y3. So y3 = 66 = (62)3, giving y = 62 = 36.

7. With y = 36 fixed, x and z are the G.P. neighbours of 36, so x · z = y2 = 1296, i.e. z = 1296 / x, and x must be a positive divisor of 1296 = 24 · 34.

8. Apply the non-zero perfect-cube condition on y − x = 36 − x to the divisors of 1296. Checking each candidate, x = 9 is the only divisor for which 36 − 9 = 27 = 33 is a (non-zero) perfect cube, giving z = 1296 / 9 = 144.

9. So x = 9, y = 36, z = 144, and x + y + z = 9 + 36 + 144 = 189.

Cross-check: 9, 36, 144 genuinely form a G.P. with common ratio 4 (9 × 4 = 36, 36 × 4 = 144); y − x = 36 − 9 = 27 = 33 is indeed a non-zero perfect cube; and substituting back, xyz = 9 × 36 × 144 = 46656 = 66, so log6(xyz) = 6 exactly — all three checks confirm the sum 189.