Express in terms of log 2 and log 3 :

$\text{log } \dfrac{75}{16} - \text{2 log } \dfrac{5}{9} + \text{log } \dfrac{32}{243}$

Simplifying the expression,

$\Rightarrow \text{log } \dfrac{75}{16} - \text{2 log } \dfrac{5}{9} + \text{log } \dfrac{32}{243} \\[1em]$

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

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

⇒ log 75 - log 16 - log 5² + log 9² + log 32 - log 243

⇒ log 75 - log 16 - log 25 + log 81 + log 32 - log 243

⇒ log 75 + log 81 + log 32 - log 16 - log 25 - log 243

⇒ log 75 + log 81 + log 32 - (log 16 + log 25 + log 243)

⇒ log (75 × 81 × 32) - log (16 × 25 × 243)

⇒ log $\dfrac{75 \times 81 \times 32}{16 \times 25 \times 243}$

⇒ log 2.

Hence, $\text{log } \dfrac{75}{16} - \text{2 log } \dfrac{5}{9} + \text{log } \dfrac{32}{243}$ = log 2.