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My current approach has been to plug in X=Z/Y in the cdf P(X<=x) , thus P(Z/Y<=x), and integrate over all values of Y. The integral I've tried to solve is ∫(0 to 1)∫(0 to Z/Y) 1 dxdy (sorry for the formatting).

This is somehow wrong, because from that integral I get z*(ln(1) - ln(0)), which is undefined (unless I'm solving the integral incorrectly, which would actually be a relief).

The correct answer is -ln(z), 0<z<1, though it's amiguous as to whether that's the cdf or pdf.

Thanks for any solutions or suggestions about where my approach is going wrong.