Question

How do I work in `log_10` without a calculator?

Answer 1

The best that you can do is to be precise with the order of magnitude unless you are able to memory the decimal values. You should know that `log` without a base means `log_(10)`.

Recall that `n=log a` is related to `10^n=a`. The integer part of `n` is the order of magnitude. So the order of magnitude for 4783 is 3. And the order of magnitude of 348734 is 5.

If you want to get a good decimal value, you will have to memorize these values for the most significant digit:
1 - 0
2 - .30
3 - .48
4 - .60
5 - .70
6 - .78
7 - .85
8 - .90
9 - .95

So estimating:

`log 4783~~3.68` using mental linear interpolation between 4 and 5
`log 348734~~5.54` using mental linear interpolation between 3 and 4