you can, when you have things downloading, a little blue downloads button appears in the bottom right hand corner of the screen
that only shows a few of them, as far as I can see. ATM I am adding, and completing, downloads without them ever showing in the bottom right, because I already have 3 large DLs listed there (which are going *very* slowly - or are stuck).
Even if the popup showed everything and worked well, a download page showing any errors in adding or downloading files (and what files were successful) seems like a no-brainer to me. I've never seen a downloader without a log before.