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Okay, so long story short, I'm a small-time sci-fi writer who likes to keep as much sci in his fi as possible. I dabble in some exponential growth equations for finding things like population growth on extrasolar colonies (y'know, as you do), and I'm currently stymied trying to find out where two functions with different growth rates would cross over in terms of t = years

The two functions essentially start at a year zero from the perspective of the fiction. One, f, starts with an initial population of 150 units and has a constant growth rate of 0.01312, while g starts with a principle of 235 and has a growth rate equal to 0.01134. By t = 294 years, f = approx. 7,100 units and g = approx. 6600 units, if my calculator math is anywhere near accurate

I apologize for my wonky format...there's certainly a good reason why I went into sci-fi and not STEM proper! Thanks in advance to anyone who can give me some pointers here

Nick Oakes
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The function of population in the $t^{th}$ year in your first planet can be represented as: $$f(x)=150(1+0.01312)^t$$ For the second planet, it is: $$g(x)=235(1+0.01134)^t$$ These formulae are basically analogous to a model of compound interest. Now, since you are interested in finding the crossover point, we have: $$150(1+0.01312)^t=235(1+0.01134)^t$$ Taking log both sides, we get: $$\ln 150+t\ln(1.01312)=\ln 235+t\ln(1.01134)$$ I trust you'll be able to plug in these values from a calculator and solve the equation.

Ritam_Dasgupta
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