I'm not sure I understand your problem.

You can open your log-file at the start of your program and just write to it with the print statements:

my $logfile = 'path/to/logfile.txt';
open(LOG, "> $logfile") or die "could not open log-file: $!";

print LOG "this is a log entry\n";
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If you need to see your log output immediately be aware of buffering issues ...

cheers

si_lence