I think that it is because of the fact that decay itself is based on probability, that cause it to be goin at a exponential rate.
For example, the half life of 1000 isotope is 10seconds.
So every 10s, the probability will kick in. Ideally, since the probability of decay is 1/2, 1/2 of the original number of isotope will remain, that is, 500.
As the next 10seconds is over, the probability will kick in again.
So this will make decay going at a exponential rate.
I personally thinks that why people in general misunderstood half life as the fact that if it takes 10seconds fo