How do I work in log_10 without a calculator?

1 Answer
Aug 21, 2014

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:

$\log 4783 \approx 3.68$ using mental linear interpolation between 4 and 5
$\log 348734 \approx 5.54$ using mental linear interpolation between 3 and 4