Cumulative distribution function help!

Hey, I'm new here. I'm stuck with this problem. Will post what I've tried to do. Hopefully someone can provide hints to help me out. Thanks.

If Z is a standard normal random variable, and Y = -ln (1-cdf(Z)), what is the distribution of the random variable Y. (note: cdf is cumulative distribution function)

Here's what I've done:

cdf(y) = P ( Y < y) = P (-ln(1-cdf(Z))<y) = ... = P ( cdf (Z) < 1 - e^-y ) -> stuck..

my approach is to find cdf of y and then differentiate it to get pdf of y, which is what's required.