Question

How do you evaluate `log424`?

Answer 1

`log 424 ~~ 2.62737`

Explanation:

Suppose we know:

`log 2 ~~ 0.30103`

`ln 10 ~~ 2.3026`

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

Then:

`log 424 = log(2*2*10*10*1.06)`

`color(white)(log 424) = log 2 + log 2 + log 10 + log 10 + log 1.06`

`color(white)(log 424) ~~ 0.30103 + 0.30103 + 1 + 1 + (ln (1 + 0.06))/ln 10`

`color(white)(log 424) ~~ 2.60206 + (ln (1 + 0.06))/2.3026`

`color(white)(log 424) ~~ 2.60206 + (0.06-(0.06)^2/2+(0.06)^3/3)/2.3026`

`color(white)(log 424) = 2.60206 + (0.06-0.0018+0.000072)/2.3026`

`color(white)(log 424) = 2.60206 + 0.058272/2.3026`

`color(white)(log 424) ~~ 2.60206 + 0.025307`

`color(white)(log 424) ~~ 2.62737`