Simplify :

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

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

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

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

= log (3 x 25) - log $(2^4)$ - 2log 5 + 2log $(3^2)$ + log $(2^5)$ - log $(3^5)$

= log 3 + log $(5^2)$ - 4log 2 - 2log 5 + 4log 3 + 5log 2 - 5log 3

= log 3 + 2log 5 - 4log 2 - 2log 5 + 4log 3 + 5log 2 - 5log 3

= (1 + 4 - 5)log 3 + (5 - 4)log 2 + (2 - 2)log 5

= 1 x log 2

= log 2

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