The value of (1/(log360) + 1/(log460) + 1/(log560)) is:

Concept: For a base and argument both greater than 1, the reciprocal of a logarithm equals the same logarithm with base and argument swapped: 1/loga(b) = logb(a). Once every term is rewritten to share one common base, the product rule logb(x) + logb(y) = logb(xy) lets several such terms be combined into one.

Application:

1. Rewrite each reciprocal term using 1/loga60 = log60a: 1/log360 = log603, 1/log460 = log604, 1/log560 = log605.

2. Add the three same-base terms using the product rule: log603 + log604 + log605 = log60(3 × 4 × 5).

3. Simplify the argument: 3 × 4 × 5 = 60, so the sum becomes log6060.

4. A logarithm of a number to its own base always equals 1, so log6060 = 1.

Cross-check: Approximating each term independently, log360 ≈ 3.73, log460 ≈ 2.96, and log560 ≈ 2.54, so the three reciprocals are about 0.268 + 0.338 + 0.394 ≈ 1.00 — confirming the exact result found above.

Result: The value of the expression is 1.