Question

How do you calculate `log 1300`?

Answer 1

`log 1300 ~~ 3.1139434`

Explanation:

Suppose you know:

`log 2 ~~ 0.30103`

`ln 10 ~~ 2.302585`

Then:

`1300 = 10*2*65 = 10*2*64*65/64 = 10*2^7*(1+1/64)`

So:

`log 1300 = log (10*2^7*(1+1/64))`

`color(white)(log 1300) = log 10 + 7 log 2 + log(1+1/64)`

`color(white)(log 1300) = log 10 + 7 log 2 + ln(1+1/64)/ln 10`

Now:

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

So:

`ln(1+1/64) = 1/64-1/(2*64^2)+1/(3*64^3)-1/(4*64^4)+...`

`color(white)(ln(1+1/64)) = 1/64-1/8192+1/786432-1/67108864+...`

`color(white)(ln(1+1/64)) ~~ 0.01550419`

and:

`log 1300 = log 10 + 7 log 2 + ln(1+1/64)/ln 10`

`color(white)(log 1300) ~~ 1 + 7*0.30103 + 0.01550419/2.302585`

`color(white)(log 1300) ~~ 3.1139434`