Mental maths: Logarithms

What is ln(8)? What about ln(100)? Or how about log₂(10)? Logarithms can help us solve questions involving exponentials or cumulative events and, whilst you will rarely have to solve them without a calculator, I think being able to is a fun trick to know. To be able to solve any logarithm you need to know 3 things:

1. The 3 rules of how logs of numbers being multiplied, divided or put to the power of work.
2. Change of base formula
3. A few logarithms off by heart (this isn’t as bad as it seems, 4-5 will do)

Let’s start with first part, the three rules. These are:

I’m not going to explain how these work as the link above will do a far better job than anything I can do.

The second part is the change of base formula:

The third part is the main ingredient and this is learning a few logarithms by heart. The reason this helps us is that using the change of base formula we can turn any logarithm into the base of our choosing; I’m going to use ln. This means that if we know our values of ln we can solve any other logarithm simply by dividing the two values together [e.g log₂(10)=ln(10)/ln(2)]. We can now get any log into ln but we still seem to need to know many values of the ln. We can solve this using the first ingredient [for example ln(10)=ln(2*5)=ln(2)+ln(5)] and looking at the example there is a clue to what ln’s are useful to learn. To minimise the number of logarithms you have to learn, it is easiest to learn the values of ln for prime numbers as you combine these to create any number you need. This means that for most practical purposes you only need to learn 5 ln values.

Using these we can form almost any ln value e.g ln(8)=ln(2³)=3ln(2) and we know ln(2). Using these three techniques you can solve any logarithm to a decent degree of accuracy, and pretty quickly. The use of using ln is also apparent when we look at the rule of 72 in another post.