Let log x = 2m - 3n and log y = 3n - 2m. Find the value of log $(x^3 \div y^2)$ in terms of m and n.

Given: log x = 2m - 3n and log y = 3n - 2m

[∵ Using nlog a = log $a^n$]

⇒ log $x^3$ = 3 log x = 3(2m - 3n)

= 6m - 9n

⇒ log $y^2$ = 2 log y = 2(3n - 2m)

= 6n - 4m

Now, log $(x^3 \div y^2)$ = log $(x^3)$ - log $(y^2)$

= 3log x - 2log y

= (6m - 9n) - (6n - 4m)

= 6m - 9n - 6n + 4m

= 6m + 4m - 9n - 6n

= 10m - 15n

Hence, the value of log $(x^3 \div y^2)$ = 10m - 15n.