Is it more sensible to log exceptions in a catch-all or in a base exception class?

Question

I'm in the process of refactoring a fairly large web app. One of the major issues is inconsistent error handling and I'm trying to come up with a sensible strategy. I've created a custom error handler, via set_error_handler that essentially turns PHP errors in ErrorExceptions and a custom base exception class, that directly inherits from Exception.

On production I'm using a generic exception catch-all, via set_exception_handler, and I'm about to add exception logging^* to the mix. My dilemma is where to do the actual logging, in the base exception class or in the catch-all.

I've thought of a couple of reasons to log it in the catch-all:

• There are quite a few exceptions in the code that need to be converted to some appropriate child of the base exception class. Until that happens, not all exceptions will get logged.
• It somehow feels more natural to do it in the catch-all, a base exception class shouldn't do more than being just that. (It might be a single responsibility principle thing, but it could just be a misguided feeling)

and one reason to log in the base exception class:

• Currently the catch-all is only used on production. It would be easy to introduce it on our other environments (development, testing) but that would call for a few adjustments, as errors are handled differently per environment, as on production they are translated to 404/503 error pages.

Is there any acceptable practice for where to logging exceptions?

_{* Logging will involve writing to a text file at first, and it may evolve to sending mails for certain types of exceptions.}

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Some clarifications, prompted by @unholysampler's answer:

I'm facing a 2*10^6 sloc codebase, with a lot of third party stuff I have no control over, and some of the code I do have control over pre-dates exceptions in PHP. And there's also some crappy recent code, we are recovering from a long period of intense pressure where we virtually had to stop thinking and just hacked.

We are actively refactoring to address all the inconsistencies and introduce a sensible error handling approach, but that's going to take some time. I'm more interested in what to do until I reach the point where errors are handled appropriately. I'll probably ask another question on a sensible exception strategy at some point.

The main motivation behind logging is to get an email on my phone whenever something bad happens on production. I don't care if data dumps get huge, if they do I'll have a cron job deleting old ones every now and then.