Evening /sci/, frenchanon here

>A little backstory : here in France when we buy chicken-related food, we have in each box one of these little magnets (pic related), each one representing a "département" of France. There are 99 of them. First when I started collected them, each one was a new one that I didn't possessed before, but then over the months, I noticed some of them were doubles of some I already own, so I wondered : can we predict of much tries (obtaining 1 new collectible randomly between all collectibles) we need for n = total number of collectibles.

So I loaded Algobox (a nice and easy-to-use program to make mathematical algorithms). User would enter the max number of collectibles (n) and how much times he wants to calculate the whole thing (this data increments when your collection is complete and the process starts all over again)

for n = the number of collectibles, and t = the number of tries needed to get the entire det of collectibles, I made a t/n = r , which hwo many times you have to multiply the number of collectibles to determine approximately the number of tries needed.

Each value had 100000 runs so it gets as precise as the program can.

For 5 objects : r ? 2.28
For 10 objects : r ? 2.93
For 50 objects : r ? 4.48
For 100 objects : r ? 5.17
For 500 objects : r ? 6.74
For 1000 objects : r ? 7.54

Looking at a graph with all points, it really looks line some sort of f(x) = a*ln(x) + b but I spent 2 hours on it and I can't find the exact formula.

How do I get the exact formula with only these numbers?