There are many ways to do this - but here is the simplest that I could figure out using rsync:
rsync -aihv --include='*/' --include='*.log*' --exclude='*' /source_folder/to/search/in/ /destination_folder/to/copy/to/
Important thing to note here is the order of include and exclude flags passed to rsync command - it is sensitive to the order in which these flags are provided.
First include flag '*/' makes rsync look into subfolders and the second flag selects all files of the format '*.log*'.
The next exclude flag excludes any file not selected as yet and then copies them over to the destination folder.
You can provide more include flags before the exclude flag to select multiple types of files as per your requirement.