ive got two suggestions for you:

Simplify

since y = log c + log x; the sd of y does not depend on the value of c.

so without loss of generality, consider the case c=1.

Use MGFs

with C=1, consider the MGF of y=logx

is just the underlying (non central) moment of X. since x follows a standard distribution, you can look this up.

Since you now have a working expression for you can use standard methods to obtain the moments of Y from . (assuming you can differenciate at t=0.)

PS: i haven't actually done this, but i assume it is tractable.

PPS: if this post makes no sense to you, then you probably have not learned "moment generating functions" yet, and youll have to solve your problem the old fashioned way.