Hello all. I am writing a fantasy novel and need to calculate population growth for realism purposes. There was another person that asked a similar question on a smaller scale, so I was plugging in the figures for my own question, but I got as far as solving for k and got stuck. Not sure how to do that. Also, I would need to know how to do this for any given year, not just 15000. Solving for k seems to be something that would be necessary each time I changed the target year, right? The equation I'm dealing with comes from the other person's question, the replies to that question bringing up the response shown below.

(Beginning of copied response with my own variables applied instead):

The population will grow according to the exponential model

P(t) = P(0)e^{kt}

P(t) = 2·2^{kt}

where t is the time in years and k is a constant.

You first need to determine k from the doubling time, 39 years.

Set t = 39 and P(t) = 4, then solve for k:

4 = 2·2^{k(39)}

(End of copied response)

4 = 2·2^{k(39)} is as far as I got. Didn't know how to proceed from there. Like I said, I know you have to solve for k. I just don't know how to do that. At risk of sounding slow, please explain as simply and with as little jargon as possible. A layman's terms explanation for how to do this is what I'm looking for. Also, this figure will be discounting any natural disasters, death rates, or any other world-issue variables. I just need a baseline figure to get started, and from there I will be able to use that figure in tandem with my own created history specific to my fantasy novel. So, again, I'm not looking for complicated answers or a discussion on world-building. I'm just looking for a baseline number to start with and the knowledge for how to get a new baseline number each time I need to change the target year. Any help anyone can give would be greatly appreciated. Thank you!