Adding and multiplying same numbers

If this were possible, then for some $m,n \in \mathbb{N}$ ,

$\begin{aligned} 12^m &= 18^n \\ \\ \left( 2^2 \cdot 3 \right)^m &= \left( 2 \cdot 3^2 \right)^n \\ \\ 2^{2m} \cdot 3^m &= 2^n \cdot 3^{2n} \\ \\ \implies \: 2m = n \; &\text{ } \& \quad m = 2n \end{aligned}$

The only solution to the resulting system is clearly $(0,0)$ which is not a valid solution given how the question is stated.

12 = 2 x 2 x 3, and 18 = 2 x 3 x 3. We need two 18's to equal the number of 2's in the 12, but we also need two 12's to equal the number of 3's in the 18. Each 18 we add will only add the number of 3's which means we need more 12's which increases the number of 2's which means we need more 18's .... and so on and so on. This is an impossible scenario.