find

!/bin/bash

DAYS=30
for dir in log_*/; do
[ ! -d "$dir" ] && continue
latest=$(find "$dir" -type f -printf '%T@\n' | sort -n | tail -1)
if [ -z "$latest" ] || (( $(echo "$latest < $(date +%s) - $((DAYS*86400))" | bc -l) )); then
echo "${dir%/}" # prints folder name without trailing /
fi
done