Logging hurts MySQL performance - but, why?
General query logs are a lot more IO than binary logs. Besides the fact that most SQL servers are 90% reads to 10% writes, the binary logs are stored in a binary format rather than plain text that uses less disk space. (How much less space? I'm not sure. Sorry.)
There are two aspects to why Apache and Exim can record every request without significant performance impact. The first is that they record the fact that a request took place but what they put in the log is usually significantly smaller than the actual request. An HTTP request is often twice as large as the line that goes in the log and even a short, plain text email is 10 or 20 times larger than the log line that accompanies it. An email with a 10MB attachment will still only have a few lines written in the log.
The second part to this is that in a normal web application there are usually dozens of SQL queries associated with a single HTTP page. Emails tend to come in even smaller numbers than HTTP requests. Your MySQL server is probably trying to log much more than either Apache or Exim.
Look at the size (uncompressed) of your MySQL binary and general logs and your Apache and Exim logs at the end of the day. I'll bet you find the MySQL general log is the largest one by a factor of at least 5.
To add to the provided answer, you'll also see a performance hit if you are logging to the same device as your MySQL data stores are residing on - if it's the same disk, you're going to be reading and writing to multiple locations all the time, slowing down the whole process.
This is true even if it's a different partition on the same physical disk.
If logging is going to a different device, that should alleviate some of the performance issues.