jmc

We can write the given equations as $$\begin{array}{rl}{\log }_{10}(A^{2}ML)& =2,\\ {\log }_{10}(R{M}^{2}L)& =3,\\ {\log }_{10}(A{R}^{2}L)& =4.\end{array}$$Then $A^{2}ML=10^2,$ $R{M}^{2}L=10^3,$ and $A{R}^{2}L=10^4.$ Multiplying these equations, we get $A^3{R}^3{M}^3{L}^3=10^9,$ so $ARML=10^3=\boxed{1000}.$