jmc

In general, if ${\log }_{12}a,{\log }_{12}b,{\log }_{12}c$ is an arithmetic progression, then we have $$2{\log }_{12}b={\log }_{12}a+{\log }_{12}c,$$or ${\log }_{12}{b}^2={\log }_{12}ac.$ Thus, $b^2=ac,$ which means that $a,b,c$ is a geometric progression.

In our case, we see that $162,x,y,z,1250$ must be a geometric progression. If $r$ is the common ratio, then we have $162r^4=1250,$ so $r^4=\frac{1250}{162}=\frac{625}{81}=\frac{5^4}{3^4},$ and $r=\frac{5}{3}.$ (Note that $x,y,z$ must be positive for the logs to be defined, so $r$ must be positive as well.) Then $$x=162\cdot \frac{5}{3}=\boxed{270}.$$