Experimental evidence for uniform distribution:

  # uniform.py

  import random

  n = 10**4
  samples = 10**6
  counts = [0] * 10

  for i in range(samples):
      random_number = random.randint(1, n)
      first_digit = int(str(random_number)[0])
      counts[first_digit] += 1

  for i, count in enumerate(counts):
      print('{} => {:.3f}'.format(i, counts[i] / samples))

Output:

  $ python3 uniform.py
  0 => 0.000
  1 => 0.111
  2 => 0.111
  3 => 0.111
  4 => 0.111
  5 => 0.111
  6 => 0.111
  7 => 0.111
  8 => 0.111
  9 => 0.111

Experimental evidence for exponential distribution:

  # exponential.py

  import random

  n = 10**4
  samples = 10**6
  counts = [0] * 10

  for i in range(samples):
      random_number = 2 ** random.randint(1, n)
      first_digit = int(str(random_number)[0])
      counts[first_digit] += 1

  for i, count in enumerate(counts):
      print('{} => {:.3f}'.format(i, counts[i] / samples))

Output:

  $ python3 exponential.py
  0 => 0.000
  1 => 0.301
  2 => 0.176
  3 => 0.125
  4 => 0.097
  5 => 0.079
  6 => 0.067
  7 => 0.058
  8 => 0.051
  9 => 0.046