Evaluate:

$\log \space \Big(\dfrac{81}{8}\Big) + 2 \log \space \Big(\dfrac{2}{3}\Big) − 3 \log \space \Big(\dfrac{3}{2}\Big) + \log \space \Big(\dfrac{3}{4}\Big)$

Given,

$\Rightarrow \log \space \Big(\dfrac{81}{8}\Big) + 2 \log \space \Big(\dfrac{2}{3}\Big) − 3 \log \space \Big(\dfrac{3}{2}\Big) + \log \space \Big(\dfrac{3}{4}\Big)$

⇒ log 81 - log 8 + 2(log 2 - log 3) - 3 (log 3 - log2) + (log 3 - log 4)

⇒ log 81 - log 8 + 2log 2 - 2log 3 - 3log 3 + 3log 2 + log 3 - log 4

⇒ log 81 - log 8 + log 2² - log 3² - log 3³ + log 2³ + log 3 - log 4

⇒ log 81 - log 8 + log 4 - log 9 - log 27 + log 8 + log 3 - log 4

⇒ log 81 + log 4 + log 8 + log 3 - log 4 - log 8 - log 9 - log 27

⇒ (log 81 + log 4 + log 8 + log 3) - (log 4 + log 8 + log 9 + log 27)

⇒ log (81 × 4 × 8 × 3) - log (8 × 9 × 27 × 4)

⇒ log (7776) - log (7776)

⇒ 0.

Hence, $\log \space \Big(\dfrac{81}{8}\Big) + 2 \log \space \Big(\dfrac{2}{3}\Big) − 3 \log \space \Big(\dfrac{3}{2}\Big) + \log \space \Big(\dfrac{3}{4}\Big)$ = 0.