Question

How do you use a calculator to evaluate the expression `log12` to four decimal places?

Answer 1

`log 12 ~~ 1.0792`

Explanation:

Since I remember (as is useful to remember):

`log 2 ~~ 0.30103`

`log 3 ~~ 0.47712125`

I can calculate:

`log 12 = log (2*2*3)`

`color(white)(log 12) = log 2 + log 2 + log 3`

`color(white)(log 12) ~~ 0.30103 + 0.30103 + 0.47712125`

`color(white)(log 12) ~~ 0.60206 + 0.47712125`

`color(white)(log 12) ~~ 1.07918125`

If you request `log 12` on a calculator, you will get a similar approximation.

If we truncated this to `4` decimal places then we would get:

`1.0791`

but the following digit is `8 >= 5`, so in order to round the value to `4` decimal places we need to round up the final digit `1` to `2` to get:

`log 12 ~~ 1.0792`

`color(white)()`
Footnote

If you remember good approximations for `log 2` and `log 3` then you can calculate approximations to `log n` for `n in { 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20,... }`, i.e. any positive integer whose only prime factors are `2`, `3` or `5`.

That seems to me to be good value for the sake of remembering a couple of numbers.