I would like to know if a key logger can be installed on a non-jail broken IPhone?
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In the practical sense, yes you can. You can use an exploit for the iPhone software to gain access to install the keylogger. If you want me to tell you how, I can't/won't.
On the nitpicky "theoretical" side - no, you cannot. If you use an exploit for the iPhone to gain access to install the keylogger, you could say that you have really jail-broken the iPhone. Thus you cannot install a keylogger on a non-jail broken iPhone.