how can one calculate the logarithmic value of any number manually.?

Answer 1

It really is a lot of work to do that. You can read about the process here:
http://mathforum.org/library/drmath/view/55566.html
I think the explanation makes use of the Taylor series expansion of the natural logarithm function around a point. You can find out about Taylor series here:
http://mathworld.wolfram.com/TaylorSeries.html
I remember reading that log tables used to take a tremendous amount of work to produce, so I suspect that the first ones were calculated using Taylor series and the properties of logarithms.

Answer 2

Here is a method to do this:

log(x) = [log (1+a)]/[log(1-a)]
= 2a[1 + (a^2)/3 + (a^4)/5 + (a^6)/7 + (a^8)/9 + ....]

provided a = (x-1)/(x+1) and x>0

So, let's say you want to find the log 10:
(10-1)/(10+1)=a=9/11

log(10)=2(9/11)[1+[(9/11)^2]/3 + [(9/11)^4]/5 +...]

If you calculate the right hand side:

(18/11)*[1+ 0.22314 + 0.089625 + 0.042855
...]

= 2.21828
Of course the more terms you calculate, the more accuracy you can obtain.

Answer 3

Solution In base 10. I will use specific example.

In order to do it off the top of your head you’ll need to memorize:

log(2) ≈ 0.3
log(3) ≈ 0.477
log(5) ≈ 0.669
log(7) ≈ 0.845

(because the others can be found using log rules and knowledge of log(2) and log(3))

Suppose you want the log(27196)

log(27196) = log( 2.7196*10^4)

by log rules we can write as:

log(10^4) + log(2.7196)

4 + log(2.7196)

Now I’ll approximate 2.7196 as 2.7 and write it as 27/10 = (3^3)/10

So now I have:

log(27196) ≈ 4 + log((3^3)/10)

and again by log rules:

log(27196) ≈ 4 + 3log(3) – 1

since I’ve memorized what log(3) is I have:

log(27196) ≈ 4 + 3*(0.477) – 1

or

log(27196) ≈ 4 .431

the art form, is making an approximation that you can work with without sacrificing too much accuracy.

Answer 4

it is simple
fisrt one logarithm table
if2345 is the number find the logof 23 to 4 mean difference 5 u get another number in 2345 it is a four digit number so in ur answer there ispoint after 3 digits (4-1)