If log (a/b) + log (b/a) = log (a + b), then:

The product rule of logarithms states that for positive m and n, log(m) + log(n) = log(mn).

1. Apply the product rule to the left-hand side: log(a/b) + log(b/a) = log[(a/b) × (b/a)].

2. Simplify the product inside the logarithm: (a/b) × (b/a) = 1, so the left-hand side becomes log(1).

3. Since log(1) = 0, log(a/b) + log(b/a) = 0.

4. The given equation states this equals log(a + b), so log(a + b) = 0, which gives a + b = 1 (again using log(1) = 0).

Cross-check: substituting a + b = 1 back into log(a + b) gives log(1) = 0, matching the simplified value of the left-hand side — confirming the relation is consistent.