Question 350855
The formula for an increase of 10% per year would be:


f = p * 1.10^n


f = future population
p = present population
n = number of years


assume the present population is 1.
assume that the population doubles.
this makes the future population = 2.


that could be 1, 1000, 1 million, 1 billion.


doesn't matter.


the future population is two times the present population, whatever the present population is.


the formula f = p * 1.10^n therefore becomes:


2 = 1 * 1.10^n


you want to find n.


if you take the log of each side of this equation, you will get:


log(2) = log(1 * 1.10^n)


there are two laws of logarithms that apply here.


the first is log(a*b) = log(a) + log(b)


the second is log(a^b) = b*log(a)


applying those laws, your formula becomes:


log(2) = log(1) + n * log(1.10)


since log(1) = 0, this formula becomes:


log(2) = n * log(1.10)


divide both sides of this equation by log(1.10) and you get:


log(2) / log(1.10) = n


use your calculator to divide log(2) by log(1.10) and you will get:


n = 7.272540897


if that's correct, then the population will double in 7.272540897 years.


plug that value in your original equation to see if that's true.


your original equation is:


2 = 1 * 1.10^n


replace n with 7.272540897 to get:


2 = 1 * 1.10^7.272540897


solve using your calculator to get:


2 = 2


this confirms the answer is correct.


the population will double in 7.272540897 years if the annual growth rate is 10%.