How do you evaluate $log 2$ ?

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Answer 1 · 2015-10-24T09:40:10

One (somewhat impractical) way is to raise to the $10$ th power repeatedly and compare with powers of $10$ as follows:

$2 < 10^1$ so $log(2) < 1$ and the portion of $log(2)$ before the decimal point is: $color(red)(0)$

If we raise $2$ to the $10$ th power, then the $log$ of the resulting number will be $10 log(2)$ , so

$10^3 = 1000 < 2^10 = 1024 < 10^4 = 10000$

So the first digit of $log(2)$ after the decimal point is: $color(red)(3)$

$1024 / (10^3) = 1.024$

To find the next decimal place, evaluate $1.024^10$ and compare it to powers of $10$ to find:

$10^0 < 1.024^10 ~~ 1.2676506 < 10^1$

So the next decimal place is: $color(red)(0)$

Then:

$10^1 < 1.2676506^10 ~~ 10.71508605 < 10^2$

So the next decimal place is: $color(red)(1)$

Divide $10.71508605$ by $10^1$ then find:

$10^0 < 1.071508605^10 ~~ 1.99506308 < 10^1$

So the next decimal place is $color(red)(0)$

$1.99506308^10 ~~ 999.002$

is just a shade under $10^3$ , so a good approximation for the next digit is: $color(red)(3)$

So $log(2) ~~ 0.30103$

How do you evaluate log 2log 2?