If log 2 = 0.3010, find the value of $\Big(\log \space \dfrac{75}{16} - 2 \log \space \dfrac{5}{9} + \log \space \dfrac{32}{243}\Big)$

Given,

$\Rightarrow \Big(\log \space \dfrac{75}{16} - 2 \log \space \dfrac{5}{9} + \log \space \dfrac{32}{243}\Big)$

⇒ log 75 - log 16 - 2 (log 5 - log 9) + log 32 - log 243

⇒ log (25 × 3) - log 16 - 2 (log 5 - log 9) + log 2⁵ - log 3⁵

⇒ log 25 + log 3 - log 2⁴ - 2log 5 + 2log 9 + 5log 2 - 5log 3

⇒ log 25 + log 3 - 4log 2 - log 5² + 2log 3² + 5log 2 - 5log 3

⇒ log 25 + log 3 - 4log 2 - log 25 + 4log 3 + 5log 2 - 5log 3

⇒ log 25 - log 25 - 4log 2 + 5log 2 + log 3 + 4log 3 - 5log 3

⇒ log 2 + log 3 + 5log 3 - 5log 3

⇒ log 2

⇒ 0.3010

Hence, $\Big(\log \space \dfrac{75}{16} - 2 \log \space \dfrac{5}{9} + \log \space \dfrac{32}{243}\Big)$ = 0.3010.