What is the best method to use?

Hello everyone,
I am no stats expert and I am just in my second year at University. I am doing a project, and I was not sure what would be the best method to come up with accurate predictions.

Here is my scenario:
I am collecting data on how many people register for a show. The registration starts about 40 days before the show. I have data on how many registration happen every day. I want to predict the approximate number of registrations at the end of the 40 days.

I had some ideas. I already have data from last year, and I compared my ideas with the data. This is what I found out.

I could use linear approximation, this would be ok towards the end, I would get close to accurate numbers, BUT even at the 20 day mark, the numbers seem very wrong!

Higher degree polynomials seem to make it worse.

Looking at the shape of the graph from last year, it looked logarithmic, and I tried fitting it with a logarithmic function, the numbers seem alright (a bit on the low side), but at some points the predicted is lower than what is the number of registrations now! (close to the 35 day mark)

What would be the best solution for my problem? I have been using excel to do the curve of best fits. I would be very happy if you could point me to some method rather than software (as I have to program this).

Thank you very much for your help!