How to find all the intersection points of the two functions log(x!) and x? I am trying to find where log(x!) and x intersect, and am unable to do so rigorously.I eventually have 2^x=x!, but I am unsure how to proceed from here. Any input as to how to go about solving this would be great.
Layton Park
Answered question
2022-11-05
How to find all the intersection points of the two functions and x? I am trying to find where and x intersect, and am unable to do so rigorously. I eventually have , but I am unsure how to proceed from here. Any input as to how to go about solving this would be great.
Answer & Explanation
jennasyliang4tr
Beginner2022-11-06Added 15 answers
If you look for a continuous function
and search for the x such that f(x)=0, I do not think that there is any analytical solution (beside the trivial solution x=0). If you plot the function, you should notice that there is a root close to and numerical method should be used, such as Newton. Starting from a "reasonable" guess , it will be updated according to
The problem is that the derivative is quite complex since
where appears the digamma function. Nevertheless, if you start at , Newton iterates will be 5.30095, 5.29033, 5.29032 which is the solution for six significant figures. If you look for a "good" approximate solution, you could replace x! by Stirling, Gosper or Burnisde approximation of the factorial. This means, that, using Gosper for example, you should solve
that is to say
This transform makes the problem much simpler. Starting at , Newton iterates will then be , , which is not too bad.