Apache does write to a large number of logs as, commonly, each website has its own traffic log:
The access_log is used to log all http requests to either the hostname of the server, requests directed at the servers IPs, or sites that resolve to the server but are no longer hosted on it.
On cPanel servers all Apache errors, regardless of site, are logged in the error_log.
In the domlogs folder each site on the server will have its own log file. These files will be the FQDN. All http traffic to a site will be logged in this log file.
Exim is the MTA (Mail Transfer Agent) that cPanel utilizes.
The exim_mainlog contains All interactions that exim handles, so both incoming and outgoing mail transactions.
The exim_rejectlog contains all connection attempts that were denied. This information is also logged in the exim_mainlog.
Cpanel does log all http traffic to WHM, webmail, and cPanel access. All cPanel logs are located in the /usr/local/cpanel/logs directory.
This access_log contains all traffic to WHM, cPanel, and webmail over http.
This error_log contains all errors that occur when accessing a cPanel related site over http or https.
Regardless of FTP deamon in use, cPanel does log connections, uploads, and downloads. However, FTP does not have its own log file it is instead threaded into the systemside messages log file.
All FTP transactions are recorded in messages. They are however interwoven with all over system messages that are logged in this file.