Why didn't I get this and why can I still not understand?

Last problem of the diff eqs test:
Show that
x^2[y''(x)] + x y'(x) + y(x) = 0

Is transformed to:
y''(z) + y(z)=0 if one makes the substitution z=ln(x).

I solved the diff eq for z, then resubbed for x to get the answer, but I can NOT figure out the steps to make the first in to the second. It is driving me mad.