use Fcntl qw(SEEK_CUR SEEK_END); #import constant

use strict;
use warnings;

use constant CLOSING_TAG => "</events>"; #deinfe the closing tag as constant

my $event1 = "<event time=\"time1\" type=\"type1\"><detail>detail1</detail></event>"; 
open(LOGFILE, "+<", "log.xml");

seek(LOGFILE, -length(CLOSING_TAG), SEEK_END);

my $cur_char;
#you may have newlines, blanks etc after the closing tag
#that's why we do the following to make the program robust
do {
   sysread(LOGFILE, $cur_char, 1);
   seek(LOGFILE, -2, SEEK_CUR) if ($cur_char ne "<"); 
} until ($cur_char eq "<");
seek(LOGFILE, -1, SEEK_CUR);

print LOGFILE $event1 . "\n"; 
#you can do as many prints as you want before close
#no more seek, we did it once and forever
print LOGFILE CLOSING_TAG;
close(LOGFILE);