But the problem with Fill in the correct answer, it changes the (ln) to (log) which is not acceptable in the input box.
Don't worry too much about it. "Fill in correct responses" is only a preview for the teacher. It doesn't change anything for students. Do not use 'Right answer' under 'Review options' in the quiz settings. Instead, make your own feedbacks.
The PHP function for the natural logarithm is "log", while in general mathematicians denote the natural logarithm by "ln". So, in the calculations you must use 'log', but with the 'Algebraic formula' answer type, you must use 'ln' to accommodate students in mathematics, physics, engineering, etc. who are more familiar with the 'ln' notation. Isn't it well done!
Another question can we insert Greek letter like \beta in the answer place holder
No, you must insert the name of a variable and the name of variables must be a string of alphanumeric characters (a-z, A-Z, 0-9 and _) that cannot begin with a number or the underscore. See MoodleFormulas.org for more details.