Consider a ASCII text file (lets say it contains code of a non-shell scripting language):
Text_File.msh:
spool on to \'$LOG_FILE_PATH/lo
If the variables you want to translate are known and limited in number, you can always do the translation yourself:
sed "s/\$LOG_FILE_PATH/$LOG_FILE_PATH/g" input > output
And also assuming the variable itself is already known