HEAVY EDIT after a few hours sleep and some coffee. Retracting my own answer and adding it to the questioon as it doesn't actually solve my problem and gave me more questions. My logic last night was that I assumed I could use our growth from 1700 to 2020 as a rough model for growth between 2093 (post Earth event) and 2435 (when the story takes place). Question has been reworded to more closely match what I need.
I am coming back to an old SciFi project based a little over 400 year from now. The background involves the decimation of Earth and mass extinction of its species and humanity save for the few that managed to escape and managed to start over on our fledgling interplanetary colonies. Then the main story takes place some 300 years later.
I'm trying to get a good feel for the expected population growth to determine a reasonable number of survivors, or find some means of hand waving the current population. Part of the story includes census data that has relevance to interplanetary politics, so I have already come up with the numbers I want. That said, if assumed population growth would be too high it is easy enough to assume tragedy or population control, but if they are too low I will need to compensate (I'm thinking long term radiation affects or rapid evolution or something increases the prevalence of plural births).
Whatever it could be, I need to know what the population count should be going in both directions.
Note: Total human population goal should be roughly 9 billion around 300 years after the decimation of Earth.
Clarifications:
Colonies exist primarily within our solar system (Venus, Mars, Titan, Europa, etc...). Habitats are slightly different on each body as they have different environments.
FTL tech was developed relatively recently (about 80 years) but it is more like crude jump gates that require a receiving end to be built via non FTL means (basically pre-fab jump gate is assembled by drones after decades of sublight travel). That makes the extra-solar colonies pretty much irrelevant for this question as they will not have more than a generation, which will likely have few births to begin with as they are startup colonies.
Retracted self-answer:
While typing up the question I thought of how I might answer it, but I would still like to know other peoples ideas on the subject or if my math is just completely wrong. It is, I didn't account for birth rate, only death rate. Bracketed to not confuse as much. Left answer here to hopefully clarify what I was originally trying for.
In 1700 there was an estimated global population of 682 million people, with a massive mortality rate that caused average age to drop below 35 years old. (this was only average because of high infant and childhood deaths, people still lived reasonably long lives if they made it to adulthood) With the current mortality rate of 7.5 deaths per 1000 people and 300 years we could use the exponential growth formula;
$y=a(1+b)^x $
where a = 682,000,000, b = (7.5/1000), and x = 300,
$y=682,000,000((1+(7.5/1000))^{300})$ I then get the result of: $y=6,416,538,709$ Or roughly 6.4 billion living humans by 2000. However, I am currently planning for a total human population of roughly 9 billion. Assuming medical science has drastically improved, but is also severely handicapped by the rigors of living in space, I would like to simplify the idea to roughly the same current mortality rate of 7.5 per 1000 people. Using the exponential decay formula; $y=a(1-b)^x $ where a = 9 billion, b = (7.5/1000), and x = 300 $y=9,000,000,000(1-(7.5/1000))^{300})$ I then get the result of: $y=940,583,032 $ Or roughly 940.6 million original survivors. Both of these scenarios seem to be off as I was hoping for around 500 million or fewer survivors. That means I would likely have to either drastically reduce the death rate or the birth rate.
