Монгол Бодлогын Сан

Заавар:\begin{gather} \log_ab=\dfrac{\log_cb}{\log_ca}\\ \log_abc=\log_ab+\log_ac\\ \log_a\dfrac{b}{c}=\log_ab-\log_ac\\ \log_{a^k}{b^{\ell}}=\dfrac{\ell}{k}\cdot\log_ab \end{gather}

Бодолт:\begin{align*} \log_{\sqrt{pq}}{\dfrac{q}{\sqrt{p}}}&=\dfrac{\log_p\frac{q}{\sqrt{p}}}{\log_p{\sqrt{pq}}}=\dfrac{\log_pq-\log_p\sqrt{p}}{\frac12\log_ppq}=\\ &=\dfrac{\log_pq-\frac12\log_pp}{\frac12\log_ppq}=\dfrac{2\log_pq-\log_pp}{\log_pp+\log_pq}=\dfrac{2\sqrt5-1}{1+\sqrt5} \end{align*}