This same code works if I ...

What is your definition of "works" ?

Your statement :

open my $read_log_file='/home/jay/my_perl_programs/log.txt';
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is not checking for errors, and ignoring any error it encounters.

The error it would get if checked is:

  No such file or directory
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This would occur even if the file does exist, because the 'open' construct not meaningful.