Country a has a birth rate of 10 per 1,000 and a death rate of 8 per 1,000. what is the doubling time for country a

You haven't said what the TIME PERIOD is during which these rates are observed.

Still, I feel like working on this one, so I'm going to assume that these birth and death rates refer to a period of ONE YEAR, and now I'll proceed to answer the question that I have invented.

If there are 10 births and 8 deaths per 1,000 EACH YEAR, then the population grows by exactly 2 per 1,000 people per year.  That's a growth rate of 0.2% per year.

In 1 year, the population grows by (1.002).  In ' X ' years, the population grows by a factor of (1.002)ˣ .

We want to know what ' x ' has to be in order for      (1.002)ˣ = 2  .

Take the log of each side:  x log(1.002) = log(2)

Divide each side by  log(1.002) :  x = log(2) / log(1.002)

x = (0.30103) / (0.0008677)

x = 346.92

Doubling time = 346.92 years

(At the end of 346 years, the population hasn't doubled yet, but they don't quite have to wait another full year. Mathematically it'll only be another 0.92 year = ~336 days.)