Here is a graph from Wikpedia showing population growth since 10,000 BCE. Note that it is virtually flat up to c2000 BCE, and again from c200 BCE to c1000 CE and then takes off (there is some evidence that it's flattening off at the moment).
Crucially it is nowhere near logarithmic.
The pattern seems to be long periods of stability/slow growth, with bursts of growth occasioned by exploitation of new territory or technological changes, e.g. agriculture, medical advances.