How to write a new entry at the beginning of a log file instead of at the end?

Shell Programming and Scripting
1

Hi Ladies and Gents,

Explanation of my question with an example:

Let's consider the script: backup_every_hour.sh

#!/bin/bash
rsync -auv $dir $backup_dir >> backup_every_hour_script.log

Each time this script is called there will be a new entry at the end of the file backup_every_hour_script.log
How do you pile up new entries at the beginning of this file?
Many thanks for your help and keep up the great work!
Cheers,

2 
logfile=backup_every_hour_script.log
{
  rsync -auv "$dir" "$backup_dir"
  cat "$logfile"
} > tmpfile
mv tmpfile "$logfile"