Depends what you mean by "songbook" and what you want to do with it....
The best tool for managing/creating PDFs is Acrobat Pro, though you need to create the text, graphics etc. in another program. Acrobat reader is available as an iOS app, though unfortunately it has the "modern" Acrobat interface which I at least find a step backards from earlier versions.
Apple's Pages app might do the job for you. Not a pdf creator, it's Apple's equivalent of Word, is available for iOS and free.
I use an app called "tunebook" quite a lot, its based on the abc notation system, but it's likely my requirements are somewhat different to yours.
Searching for "ios songbook app" throws up quite a lot of hits. You might find this site useful -
http://www.stevemacdonald.org/tablet/apps.html