Observing that the logarithm of a product of two elliptic operators differs from the sum of the logarithms by a finite sum of operator brackets, we infer that regularised traces of this difference are local as finite sums of noncommutative residues. From an explicit local formula for such regularised traces, we derive an explicit local formula for the multiplicative anomaly of ζ-determinants which sheds light on its locality and yields back previously known results.

pseudodifferential operators, noncommutative residue, canonical and weighted traces, zeta and weighted determinants