How to manipulate logarithms to solve problems in the real world

By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

31 March 2024

In last week’s article, we saw how logarithms come into the question of how long it would take to fill up the earth with people. In summary: growing at the rate of 0.15 per cent per month, how many months does it take for the world population to grow from the current 8.1 billion to 624 trillion, that is,77,037 times. The answer is the logarithm of 77,037 in base 1.0015.

Now, we normally read out logarithms from mathematical tables, but, unfortunately, there no table in base 1.0015! They are mostly available in base 10.  So, we need to do some manipulation.

If we multiply 3 by itself six times, that is, 3x3x3x3x3x3, we get 729. Therefore, the logarithm of 729 in base 3 is 6. We can split the multiplication into two parts, thus: 3x3x3x3=81 and 3x3=9. Then multiply the two products: 81x9=729.

We note that the logarithm of 81 in base 3 is 4 and the logarithm of 9 in the same base is 2. We immediately notice that 4+2=6; therefore, it turns out that the logarithm of the product of two numbers. In mathematical lingo: since 81x9 =729, then log81 + log9 = log729. Notice that the base of the logarithm doesn’t matter!

Let’s take this one step farther: since 9 = 3x3, then log9 = log3+log3; in other words, log9 = 2xlog3. We can turn this around and write it as log9 divided by log3 = 2. Now this is a good place to stop and take a cup of tea!

We are now ready to tackle the original problem: we know that 1.0015 raised to a certain power should give 77,037. Therefore, if we find the logarithm of 1.0015 in base ten and multiply it by this unknown power, the result should be equal to the logarithm of 77,037.

Consequently, the unknown power – which is the number of months – is simply the logarithm of 77,037 (=4.8867) divided by the logarithm of 1.0015 (=0.000651). When we divide 4.8867 by 0.000651, we get 7,506 months. That is, about 625 years.

Admittedly, the above looks like too much calculation for an obviously unrealistic situation – that no deaths occur in a period of six centuries. However, it is good practice to illustrate how one can solve a real problem; for example, if you invest money at 15 per cent per annum, how long will it take for your investment to double in value? That’s a real problem from the real world and the calculation is exactly the same as the one above.