It's kind of like hardcore paper mache. To get a solid log, you must change the structure of the newspaper. That is from sheets to pulp. Usually there's no shortcuts in paper mache, but you'll learn how to make these logs very quickly using a rubber mallet.
The paper logs burn best when combined with wood. They will create more ash than wood. The hole through the middle will help it to burn and make the drying out process quicker.
Step 1: Prepare the newspapers
If you have a leaky gutter with a bucket under the drip or collect rainwater in a tank it would be a great idea to use rain water for the logs.
The newspapers in this instructable were soaked for two days. One day is enough. If you want to speed up the softening process put a squirt of cheap detergent in the water.
wingbatwu suggested adding flour to the water which sounds like a really good way of helping everything bind together.