I think it might be a matter of there not being many habitable worlds in the galaxy. Like i said with earth's not growing on trees, most of the extra-solar planets we've identified are down-right inhospitable to life as we know it. Lots of dead rocks and gas giants. Maybe you could terraform them, but even if you have the technology that takes a lot of work, so you wouldn't colonize freely, more on a infrequent basis. You also have the problem of colonization being slowed by contact with other advanced life. And in that example, if you were limited to traveling at a speed less than light, it's going to take a number of years to get there/send later supplies. Which is dwarfed by the amount of time the galaxy has existed for, yes.
But time isn't the only variable. The harder it is to do something, the less willing people are to try unless there's a sweet payoff. And it's sounding like colonization is pretty hard.
Not to mention is that colonization vessels that travel at sub-light would have to be HUGE. Packed with people, supplies, fuel, etc. Exponential growth doesn't work on this since a colonization ship is something that would have to built at a large scale shipyard, something that probably is only at the homeworld and nearby worlds. They're not something that would be built at every colony since building up the industry to create and maintain a shipyard would be a large scale project, and it makes sense to keep things specialized too. Not every city in the US has farmland and factories. We have farming areas of the country along with manufacturing areas.
So you'd really have colonization speed decreasing with each world, as the worlds you try to travel to become farther away from the homeworld, which makes it longer to travel,more costly (need more supplies), and higher risk. Additionally, all of these things make people less enthusiastic to go in the first place.
EDIT: Just to add on, your astronomy professor's argument assumes that each colony would become a copy of the homeworld, matching it in population and industry. Which just isn't going to happen unless breeding rates are out of control, and you're REALLY good at quickly building up all the infrastructure to support that expanding population.