AMC 12 2013a Problem 14

The sequence

$\log_{12}{162}$, $\log_{12}{x}$, $\log_{12}{y}$, $\log_{12}{z}$, $\log_{12}{1250}$

is an arithmetic progression. What is $x$?

\textbf{(A)} \ 125\sqrt{3} \textbf{(B)} \ 270 \textbf{(C)} \ 162\sqrt{5} \textbf{(D)} \ 434 \textbf{(E)} \ 225\sqrt{6}