Thanks for giving such a nice explanation @martixingwei!

Basically, the formula of year n's population is p0*(1-percent)^n+aug*[(1+percent)^(n-1)+(1+percent)^(n-2)+...+1], which is also equal to p0*(1-percent)^n+aug*[(1+percent)^n-1]/percent, then p0*(1-percent)^n+aug/percent*(1+percent)^n-aug/percent and finaly (p0+aug/percent)*(1+percent)^n-aug/percent.
By solving p = (p0+aug/percent)*(1+percent)^n-aug/percent for n, we can get n = log( (p+aug/percent)/(p0+aug/percent), 1+percent).

In lechevalier's solution, his percent is actually 1+percent and his r is -aug/percent, so his last line became log((p-r)/(p0-r), percent).

Happy to hear that!

Instead of multiplying by a percentage until population exceeds target, you can directly use log(..., percentage) to calculate the result you want.