Well one shorter would be to use grep instead of the foreach @cont.

my @contents = grep { /^[Bb].+.txt$/ } @cont;
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Also you probably want your RE to be:

/^[Bb].+\.txt$/
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So that is only matches a file ending in .txt rather than the current match which would match 'Bobtxt' or 'Bob.txt'

Also I don't see any point in opening the log files since you are only printing the name out anyway. You can remove the open and close for a big speedup. If you want to print the contents of the logfiles you might try:

open(FILE, "<$logfile"); # specify read explicitly with <
print "$logfile:\n";
print while(<FILE>);
close(FILE);
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This will print the name of the file followed by its contents.

Update: Also the [B|b] is not desired as it tries to match B or | or b and you probably just want B or b which the RE's provided will match. The | character in a grouping (not a char class) provides a way to alternate groups of characters like (abc|def)ghi which would match abcghi or defghi but not aecghi etc.