Actually on a long enough scale population growth
is exponential and has been increasing at a phenomenal rate (by such time scales). However the good news is that the time it takes for the world population to double is estimated to increase. (Ed, note that any growth on any scale can be read as exponential. On a geological scale the human population growth is akin to the big-bang, but on a daily scale it's equivalent to a lethargic snail.)
This is due to the fact that if current trends continue, it will have taken 12.3 years for the world population to go up by a billion (however it should be noted, earlier estimates -1990 IIRC- suggested it should have hit 10 billion ~2012), which is a mild increase over the last billion of 12.25 years. I believe the next billions are expected to take 13, 15 and then IIRC 18 years.
There's many relevant factors effecting population growth. Many of the diseases that plagued our ancestors are almost completely cured in the west (I've never heard of someone dying from a broken arm, however 200 years ago it was a prominent risk).
We also have food preservation techniques that no one had before. Everywhere from penicillium being used in meats to prevent bacterial growth to refrigeration, vacuum sealing, freeze drying and even vacuum freezing (where the moisture is literally sucked out of food, meaning it can last unrefrigerated for up to 10 years). There's also irradiation techniques that, given proper storage, can enable food to last indefinitely.
There are also vast misconceptions about over population that plague these discussions. For example, China is less densely populated than the UK and Japan, yet has food struggles because they don't have the economy to import food on mass. Many African countries plague by starvation are desolate compared to most countries, yet neighbouring countries have populations almost one hundred times as populated (note that this even applies in the landlocked countries interior and isn't a comparison between a country with a large ocean border and one with no border).