2013 AMC 12A 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} \qquad \textbf{(B)} \ 270 \qquad \textbf{(C)} \ 162\sqrt{5} \qquad \textbf{(D)} \ 434 \qquad \textbf{(E)} \ 225\sqrt{6}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.