Question

How do you find the value of log(base ten) 2 without calculator,,,? thanks for the information

Answer 1

`log 2 ~~ 0.30103`

Explanation:

Usually when you want to find a non-trivial logarithm base `10` without the use of a calculator you will use a few known values, such as:

`ln 10 ~~ 2.302585092994`

`log 2 ~~ 0.301029995664`

`log 3 ~~ 0.477121254720`

Also we might commonly use the series for `ln(1+x)`...

`ln (1+x) = x-x^2/2+x^3/3-x^4/4+...`

Note that one of the common values given is often `log 2 ~~ 0.30103`.

How else might we find it?

First notice that:

`2^10 = 1024 ~~ 1000 = 10^3`

What does that mean in terms of common logarithms?

`log 2^10 = 10 log 2 ~~ log(10^3) = 3`

Essentially what happens when you raise a number to the `10`th power is that its logarithm (in any base) is multiplied by `10`.

So we could write:

`log 2 = 1/10 log 2^10`

`color(white)(log 2) = 1/10 log (10^3 * 2^10/10^3)`

`color(white)(log 2) = 1/10 (log 10^3 + log (2^10/10^3))`

`color(white)(log 2) = 1/10 (3 + log 1.024)`

`color(white)(log 2) = 0.3 + 1/10 log (1 + 0.024)`

`color(white)(log 2) = 0.3 + 1/(10 ln 10) ln (1 + 0.024)`

We can approximate `ln(1+0.024)` using the series:

`ln(1 + 0.024) ~~ 0.024-(0.024)^2/2+(0.024)^3/3`

`color(white)(ln(1 + 0.024)) ~~ 0.024-0.000576/2+0.000013824/3`

`color(white)(ln(1 + 0.024)) ~~ 0.024-0.000288+0.000004608`

`color(white)(ln(1 + 0.024)) ~~ 0.023716608`

So:

`log 2 ~~ 0.3+0.023716608/23.02585092994`

`color(white)(log 2) ~~ 0.3+(0.023025851+0.000690757)/23.025851`

`color(white)(log 2) ~~ 0.3+0.0103`

`color(white)(log 2) ~~ 0.30103`