Can a professor grade students based on what she "remembers" with no proper record? I was graded unfairly in a communication class. I participated more than some students yet the professor said she couldn't remember me speaking out in class. I asked her if she has any record on that and she said it would be hard for her to record if everyone speaks out

Some of the students who were less active were given better marks than those who were more active, it was inconsistent. 
There were 23 students in the class and 13 classes in total. By the end of the class, she could not even remember the names and faces of each and everyone. 
How should I approach this?
Check if your school has a grade appeal process.

Many schools in the US have one. It will bring in some other administrators (e.g., chair of the department or dean) to review your argument and assess the professor's rationale for giving you the grade.

You might have a good chance with this since she has no written justification for your grade or other student's grades and relies only on her memory. Additionally, if you can remember what you spoke about during the class on specific days (and the students who spoke immediately before/after you), you might want to go ahead and write that down so that you can bring it up with your appeal.