A week or so ago I asked why $ \prod_{n\geq 1}\frac{1}{(1-x^n)^{m_n(q)}}=\frac{1}{1-qx} $ where $m_n(q)$ the number of irreducible monic polynomials with degree $n$ over the finite field of order $q$.
Why does taking logarithms then imply $ \sum_{n\mid r}nm_n(q)=q^r $ for any $r$? I know taking logarithms will change it to an additive identity, but don't see how this particular equality falls out. Thank you.