Simplify: 1/logxy(xyz) + 1/logyz(xyz) + 1/logzx(xyz)

Concept: The change-of-base identity states 1/logb(a) = loga(b) for any valid bases. Also, logarithms with the same base add via loga(m) + loga(n) = loga(mn), and loga(a) = 1.

1. Rewrite each term using 1/logb(a) = loga(b): 1/logxy(xyz) = logxyz(xy), 1/logyz(xyz) = logxyz(yz), 1/logzx(xyz) = logxyz(zx).

2. Add the three logarithms (same base xyz): logxyz(xy) + logxyz(yz) + logxyz(zx) = logxyz(xy · yz · zx).

3. Simplify the product inside the logarithm: xy · yz · zx = x2y2z2 = (xyz)2.

4. So the expression becomes logxyz((xyz)2) = 2·logxyz(xyz) = 2·1 = 2, using loga(mn) = n·loga(m) and loga(a) = 1.

Cross-check: Take x = y = z = 10, so xyz = 1000 and xy = yz = zx = 100. Then log100(1000) = 3/2, so each original term 1/log100(1000) = 2/3, and the sum of three such terms is 3 × 2/3 = 2 — matching the result above.