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 \[x = 162 \cdot \frac{5}{3} = \boxed{270}.\]