Linux shell file. mkdir and pushd commands not doing what I'd like

I'm trying to create a directory if it does not exist, and the further down the line, create a log file in that directory. It would be in the current users home directory
In pseudocode:

If directory "/home/LOGS" does not exist
make directory "/home/LOGS"

Then whenever a log file is created, it will save in that directory, not the directory that the program is being run from.
This is what I have so far, and it's just not working...

pushd -n  $"../LOGS" >/dev/null 2>&1 

#Create the logs folder
if [ ! -d $"home/LOGS" ]; then
 mkdir -p $"home/LOGS"
fi 

If it helps, my files are being saved like so:

     function savefile {
# Save to file
  if [ -e $username.log ]; then
    echo "Username already exists, returning you to the menu."
    sleep 2
    clear
      menu
  else
    echo $fullname >> $username.log
    echo $password >> $username.log
      curdate=$(date +'%d/%m/%Y %H:%M:%S')
    echo $curdate >> $username.log
  echo "Creating your account"
    sleep 2
    clear
  echo "Account created"
  echo
    afterBasic
  fi 
}

Any help would be greatly appreciated.