Logarithms and how I calculated 625 years to fill the earth

By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

24 March 2024

A regular reader who wants to remain anonymous has asked me to explain in detail how I calculated the time it would take to fill the earth with people (Sunday Nation, March 10, 2024). He is particularly interested in knowing how logarithms feature in the calculation.

As a reminder, we found that it would take 624 trillion people to completely fill up the land surface of the earth. We also noted that about 24.5 million babies had been born in the world since the beginning of the year and that the world population was about 8.1 billion people.

The first step in the calculation is to find out what proportion the new-borns are in the population. That is, 24.5 million divided by 8.1 billion., this comes to about 0.003, or about 0.3 per cent. Since we were doing the working about two months into the year, we may divide this result by two to get the monthly birth rate. That’s how I got 0.15 per cent.

In other words; assuming no deaths occur, the world population would grow by a factor of 1.0015 every month. So, the question becomes: if we start with 8.1 billion and increment this by 1.0015 monthly, how many months would it take to reach 624 trillion?

To answer that one, we need to know the ratio of the final population to the current one. That is: what is 624 trillion divided by 8.1 billion? The answer is 77,037. Now we can rephrase the question, thus: how many times do we need to multiply 1.0015 by itself to get 77,037? That number of times is simply the number of months, but it is not a straightforward calculation: it calls for knowledge of logarithms.

I explained the meaning of logarithm in this column in June 2011. Admittedly, that’s a very long time ago, so let me go over it again. We know that 2 x 2 x 2 = 8; that is, 2 raised to the power of 3 is equal to 8. Therefore, we can restate this as: “the logarithm of 8 in base 2 is 3. Get it?”. Similarly; 10 x 10 x 10 x 10 =10,000; therefore, the logarithm of 10,000 in base 10 is 4.

So, our population question becomes: what is the logarithm of 77,037 in base 1.0015? That calculation is also not straightforward! I will demonstrate how to solve it in the next article.