To inject a Php payload in the log file of a server, just send an HTTP GET request containing php code in the url.
For an ssh or ftp server, inject the payload in login. Use then an LFI on the log file to trigger the payload.

Usual log files locations:
Apache
/var/log/apache/access.log
/var/log/apache/error.log
/usr/local/apache/log/error_log
/usr/local/apache2/log/error_log
/var/log/httpd/error_log

Nginx
/var/log/nginx/access.log
/var/log/nginx/error.log

Ssh
/var/log/sshd.log

Log files location can be found in webservers config files:
Nginx: /etc/nginx/nginx.conf
Look for : access_log /spool/logs/nginx-access.log