The "expected" data set is the linear fit. The following script uses linear regression and the R^2 metrics to calculate a measure of fit for your three datasets:

use strict;
use warnings;
use Statistics::LineFit;

sub fit {
  my $fit = Statistics::LineFit->new();
  $fit->setData( @_ );
  return $fit->rSquared();
}

my @data = ( [ qw( 5 5 34 44 114 169 177 184 270 339 361 364 442 511 5
+30 554 555 587 709 709 735 778 791 859 871 899 903 926 933 952 ) ],
         [ 0.5, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 
+8191, 16383, 32767, 65535, 131071, 262143, 524287, 
           1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 671
+08863, 134217727, 268435455, 536870911, 1073741823 ],
         [ 1.713125e-005, 1.748086e-006, 2.101463e-006, 1.977405e-006,
+ 3.597675e-006, 3.725492e-006, 3.924736e-006, 2.902199e-006, 
           3.988645e-006, 8.210367e-006, 3.360837e-006, 5.202907e-006,
+ 7.082570e-006, 8.778026e-006, 7.079562e-005, 9.100576e-005, 
           5.258545e-005, 9.292677e-005, 1.789815e-004, 2.113948e-003,
+ 7.229146e-004, 1.428995e-003, 2.742045e-003, 5.552746e-003, 
           1.822390e-002, 2.220999e-002, 4.316067e-002, 8.876963e-002,
+ 1.751072e-001, 3.494051e-001, 7.155960e-001, 1.347822e+000 ] );

print "    linear  loglinear     loglog\n";
for my $d (@data) {
  my @x = 1..@$d;
  my @logx = map log, @x;
  my @logd = map log, @$d;
  printf "%10.2f %10.2f %10.2f\n", fit( \@x, $d), fit( \@x, \@logd), f
+it( \@logx, \@logd );
}
[download]

The result is

    linear  loglinear     loglog
      0.99       0.69       0.95
      0.26       1.00       0.86
      0.26       0.90       0.58
[download]

which shows that the first data set describes a linear relationship while the others are more of log type (the largest R^2 wins). If you have a stats package at hand (or even Excel only) you can do the same thing and visualize the results.