I'm wondering what kind of application you write. A virus or a Trojan horse? Anyway ...
Encrypt each entry alone, convert it to some string (Base64, for example) and then log that string as the "message".
This allows you to keep parts of the file readable and only encrypt important parts.
Notice that there is another side to this coin: If you create a fully encrypted file and ask the user for it, she can't know what you will learn from the file. Therefore, you should encrypt as little as possible (passwords, IP addresses, costumer data) to make it possible for the legal department to verify what data is leaving.
A much better approach would be to an obfuscator for the log file. That simply replaces certain patterns with "XXX". You can still see what happened and when you need a specific piece of data, you can ask for that.
[EDIT] This story has more implications that you'd think at first glance. This effectively means that a user can't see what's in the file. "User" doesn't necessarily include "cracker". A cracker will concentrate on encrypted files (since they are probably more important). That's the reason for the old saying: As soon as someone gets access to the machine, there is no way to prevent him to do anything on it. Or to say it another way: Just because you don't know how doesn't mean someone else also doesn't. If you think you have nothing to hide, you haven't thought about yourself.
Also, there is the issue of liability. Say, some data leaks on the Internet after you get a copy of the logs. Since the user has no idea what is in the log files, how can you prove in court that you weren't the leak? Bosses could ask for the log files to monitor their pawns, asking to have it encoded so the peasants can't notice and whine about it (or sue, the scum!).
Or look at it from a completely different angle: If there was no log file, no one could abuse it. How about enabling debugging only in case of an emergency? I've configured log4j to keep the last 200 log messages in a buffer. If an ERROR is logged, I dump the 200 messages to the log. Rationale: I really don't care what happens during the day. I only care for bugs. Using JMX, it's simple to set the debug level to ERROR and lower it remotely at runtime when you need more details.