heights of towers and titchy little numbers up and to the right…(logarithms and indices)

OK so this is worth three threes, 3 x 3


The ‘answer’ or product is 9

Another way of representing the same meaning is:

3x3 log2

It means 3 raised to the power 2

The height of the tower is 2.

Next its 3 x 3 x 3 followed by  3 x 3 x 3 x 3:


3 x 3 x 3, answer is 27

3x3x3 log3

This means the same, 3 x 3 x 3, 27


3 x 3 x 3 x 3, answer is 81

3x3x3x3 log 4

This means the same, 3 x 3 x 3 x 3,  81

There is another common way of writing these power towers down:

powers of 3-b

The little number is THE HEIGHT OF THE TOWER, called the INDEX

Just as the names of the numbers after ten have unusual names like ‘eleven’ and twelve’, the first two of these have other common names. Numbers raised to the power 2 can be called ‘squared’, i.e. 3 squared, and numbers raised to the power 3 can be called  ‘cubed’. After that its all ‘regular’, they are called something ‘raised to the power something’, such as 3 raised to the power 4 in the last example.

( You can easily make a square using three threes on the flat. Multiply the length of 2 sides together and you get 9, the same as 3 x 3 . You can make a cube of three rods, then if you multiply the 3 sides together you get 3 x 3 x 3 which is 27.

By the way, the square root of 9 is 3, and the cube root of 27 is 3…its all playing in the same ball park).

10 raised to the power 100 is called googol.

The number of atoms in the universe is about 10 raised to the power 79, so googol is far greater than the number of atoms in the whole universe.


powers of 3-b

So the INDICES are the little numbers. Now wait for it,

THEY ARE ALSO THE LOGARITHMS OF THE ANSWER, provided you are talking about the same colour rods, in this case 3 rods or light green:

2 is the LOGARITHM of 9 in BASE 3

3 is the LOGARITHM of 27 in BASE 3

4 is the LOGARITHM of 81 in BASE 3

The main use for these logarithms in the olden days of torture in school was that:


The same is true of indices, and these days this is still useful with very large or small numbers, as in maths, physics and engineering etc.

I’m getting a bit fed up with this now, so just let’s say to finish and for a quick example, 3 squared multiplied by 3 raised to the power 200 is 3 raised to the power 202. As to the answer, I’m too tired to figure it out. Goodbye.

the LOGARITHM is the height of the tower

Logarithms sound bad I know, but they’re not.

Look, this is 10×10

Say, ‘ten cross ten’. The answer is 100 is it not? It is 2 layers high:

10x10 log2

This means the same thing, but we use less rods when the towers get higher. We can turn this one on its side if we like:

10x10 log 2 flat

Now it looks rather like this flat representation:10 power 2

This means 10×10, 10 squared, 10 raised to the power 2, answer is 100


…if you are using orange rods and you call an orange rod 10


The only thing to stress is that the tower must be made of the same colour of rods.

This colour is called the BASE