Let \(m = \log_bp\) and \(n = \log_bq\).
Then \(b^m = p\) and \(b^n = q\), so \(\dfrac{p}{q} = b^{m - n}\), which is equivalent to \(m - n = \log_b \dfrac{p}{q} \). Replacing \(m\) and \(n\) yields \(\log_b p - \log_b q = \log_b \dfrac{p}{q}\).