In my previous post I described how I calculated the layout of the logarithmic scales for the slide rule watch. I used a computer to recreate this pre-computer aged device. You could consider this method ironic – or cheating!

When slide rules were originally developed, the scales would have had to have been manually calculated. How could we get these results with pencil and paper?

Here is a method that works using only simple arithmetic. It can be done on (lots of) paper. You can also try it with a simple calculator to save time.

How to calculate the log of 3.

The result will be 0.???? since it will be less than 1

### Calculate the first decimal place:

Multiply 3 by itself 10 times to get 59,049

Count the number of digits on the left of the decimal and subtract 1 to get 4

The log of 3 is now known to be 0.4???

### Calculate the second decimal place:

Move the decimal place in the result above so as to have one digit to the left of the decimal place to get 5.9049

Multiply 5.9049 by itself 10 times to get 51,537,752.07320

Count the number of digits on the left of the decimal and subtract 1 to get 7

The log of 3 is now known to be 0.47??

### Calculate the third decimal place:

Move the decimal place in the result above so as to have one digit to the left of the decimal place to get 5.153775207320

Multiply 5.153775207320 by itself 10 times to get 13,220,708.19481

Count the number of digits on the left of the decimal and subtract 1 to get 7

The log of 3 is now known to be 0.477?

Calculate the forth decimal place:

Move the decimal place in the result above so as to have one digit to the left of the decimal place to get 1.32207

Multiply 1.32207 by itself 10 times to get 16.313502

Count the number of digits on the left of the decimal and subtract 1 to get 1

The log of 3 is now known to be 0.4771

you can continue to do this for as many decimal places as required.

To be honest, I am not fully clear on the mathematical underpinnings that make this method work! I’ve tried it for several numbers and it corresponds with what I get out of Excel.