I didn't hear about any software that supports your needs. But if you are programmer, shou should understand how to write this in Perl.
Here i found example how to get all file. This isn't your resolve, but it help you understand how to do:
another, help for library Net::FTP :
using this you can open ftp connection to server (first example)
make authorization (first example)
second URL help you how to get last part of file using methods:
->ascii set transfer mode to ascii because this is logfile
->size ( FILE ) get size of file for counting where you should start
->restart ( WHERE ) set file coursor where from in the file you want start reading
->read ( BUFFER, SIZE [, TIMEOUT ] ) directly read small SIZE part into BUFFER variable
length(BUFFER) tell you how much data you get. If it's ZERO, you propably reached EOF
print ("$BUFFER\n") simple print buffer content to the screen
Notice, this is not completly resolution but a way how to do.
Only one reason why I write this all is there probably are no software ready for your task.
If you don't know how to write that program, simply ask on stackoverflow.com , not superuser. There are many programmers that are ready for help.
If it's possible, check your ftp site supports download resuming. Whithout this reading end part of file is impossible.
Another partial resolution may be log file rotation, one file a day or a hour. this depends how fast log file grows.