Express $\log_{10} \space \Big(\dfrac{\sqrt{pq^3}}{r^2s}\Big)$ in terms of log₁₀p, log₁₀q, log₁₀r and log₁₀s.

Given,

$\Rightarrow \log${10} \space \Big(\dfrac{\sqrt{pq^3}}{r^2s}\Big) \\[1em] \Rightarrow \log{10} \space \sqrt{pq^3} - \log{10} \space {r^2s} \\[1em] \Rightarrow \log{10} \space ({pq^3})^{\dfrac{1}{2}} - (\log{10} \space {r^2} + \log{10} \space {s}) \\[1em] \Rightarrow \dfrac{1}{2}\log{10} \space {pq^3} - (2\log{10} \space {r} + \log{10} \space {s}) \\[1em] \Rightarrow \dfrac{1}{2}(\log{10} \space {p} + \log{10} \space {q^3}) - 2\log{10} \space {r} - \log{10} \space {s} \\[1em] \Rightarrow \dfrac{1}{2}(\log{10} \space {p} + 3\log{10} \space {q}) - 2\log{10} \space {r} - \log_{10} \space {s} \\[1em]

Hence, $\log${10} \space \Big(\dfrac{\sqrt{pq^3}}{r^2s}\Big) = \dfrac{1}{2}(\log{10} \space {p} + 3\log{10} \space {q}) - 2\log{10} \space {r} - \log_{10} \space {s}.