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 for half the isotope to decay, it will take another 10seconds for the rest of the isotope to decay, is because they view it as it takes 10seconds for 500 isotopes to decay. Therefore it will take another 10seconds for the rest to decay. They are not linear. Decay is a probability as I said above.