United Nations estimates are that the
world population will increase
from 6 to 9 billions in the first half of this century.
but the increase in population will vary
substantially between countries.
How do we work out the
extent to which population will increase
over a period of years in any one country?
There are all kinds of sophisticated ways of doing this,
by looking at birth rates, death rates, immigration, and so on
but a simpler way is to just assume that
the current increase in population
will continue indefinitely.
As an example, let's take Mexico.
Its population, currently around a hundred million, is growing faster than some places but slower than others.
If we assume an annual increase of two per cent
how many years will it take the
population to double?
We need a basic grasp of logs to find the answer.
The problem we're trying to solve is this.
How long will it take for a country's population to double
if it's growing at 2 percent per annum?
The country's future population is the
current population
multiplied by k to the t
where k is the scale factor
and t is the time period
in our case, the number of years.
Here we're interested in finding t, the number of years that it takes for the population to double.
So the current population is 1; we want
to know, when does it reach 2?
i.e. when does it double?
So for this limited purpose we don't need
to know the size of the population
only the amount to time that it will
take any population
to double in size.
So our formula is that 2 = k to the t
and we want to find t.
Solving for a power requires the use of logs.
We're using natural logs
and we know that the log of 2 is equal to
the log of k to the t.
But from our log law
log a to the b is equal to b log a.
Remember, that's true of any log
of any base, not just natural logs.
So the log of 2 equals t log k
and we need to rearrange that expression
so that we can discover t, the unknown.
We want t on the left hand side of the
equation
so rearranging, we have
t equals log of 2 over log of k.
Now we can find t,
the amount of time it takes for the
population to double.
Here the growth rate is 2 percent
so the scale factor is 1.02.
Log 1.02 equals 0.0198
Log 2 equals 0.6931.
So substituting into the formula we have
t= 0.6931 over 0.0198
So, solving for t,
t equals 35.
So we have a seemingly modest growth
rate of population
but because of the nature of compounding, a 2% annual growth rate
leads to a doubling of population in
just 35 years.
