WordPress, Drupal and Joomla are all good choices for running a blog if you choose to use a dedicated hosting service (e.g. hostclear, etc.), and they're all extremely well supported by their dedicated community.
Here's a good article which offers a high level comparison of the three:
http://www.goodwebpractices.com/other/wordpress-vs-joomla-vs-drupal.html
Speaking from personal experience, I think it's worth getting a decent hosting service like HostClear, they're pretty cheap these days and gives you domain registration for free too. I've been running my blog for about 8 months now, at less than £3 a month and I've been more than happy with the service I got! And having tried both Joomla and WordPress, I found WordPress much easier to get started with.