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Taylor Francis
science forum beginner

Joined: 18 Jul 2006
Posts: 1 Posted: Tue Jul 18, 2006 2:43 pm    Post subject: Logarithm Does anyone know a formula for calculating a logarithm by hand? Frederick Williams

Joined: 19 Nov 2005
Posts: 97 Posted: Tue Jul 18, 2006 2:50 pm    Post subject: Re: Logarithm Taylor Francis wrote:
 Quote: Does anyone know a formula for calculating a logarithm by hand?

What base?

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Joined: 02 May 2005
Posts: 603 Posted: Tue Jul 18, 2006 7:04 pm    Post subject: Re: Logarithm On Tue, 18 Jul 2006 09:43:14 -0500, Taylor Francis <radiulo@gmail.com>
wrote:

 Quote: Does anyone know a formula for calculating a logarithm by hand? Pontypridd Chess Club (Da
science forum beginner

Joined: 18 Jul 2006
Posts: 1 Posted: Tue Jul 18, 2006 8:47 pm    Post subject: Re: Logarithm For finding logs to the base e there are a couple of formulae depending on
the size of x. For example, provided x is greater than 0.5 the following
series could be used:

(x-1) / x + (x-1)^2 / 2x^2 + (x-1)^3 / 3x^3 ...

Unfortunately, the larger x is the more terms it takes to find a reasonable
approximation.

"Taylor Francis" <radiulo@gmail.com> wrote in message
> Does anyone know a formula for calculating a logarithm by hand? Mike1161

Joined: 14 Oct 2005
Posts: 50 Posted: Wed Jul 19, 2006 12:39 am    Post subject: Re: Logarithm Quote: Does anyone know a formula for calculating a logarithm by hand?

To calculate n = log[base k] of X.

Step 1:
Set n = 0. Set p = 1

Step 2
While X > k do the following
Divide X by k
Increment n

Step 3
Repeat
Set k to sqrt(k)
Divide p by 2
if X > k then do the following
Divide X by k
Add p to n
until p < acceptable error

n will be the required logarithm.

For example:
Lets say we want to know the value of log2(19) = n.

n X k p
0 19 2 1 <- step 1 complete
1 9.5 2 1
2 4.75 2 1
3 2.375 2 1
4 1.1875 2 1 <- step 2 complete
4 1.1875 1.414214 0.5
4 1.1875 1.189207 0.25
4.125 1.088942 1.090508 0.125
4.1875 1.042775 1.044274 0.0625
.... ... ... ...

at which point we know the log{base 2] of 19 is approximately 4.1875 to an accuracy of 0.0625.
So all it involves is addition, division and sqrt calculation, all of which can be done easily with a pencil and paper.

Mike alex.lupas@gmail.com
science forum beginner

Joined: 23 Feb 2006
Posts: 47 Posted: Thu Jul 20, 2006 5:17 am    Post subject: Re: Logarithm Taylor Francis wrote:
 Quote: Does anyone know a formula for calculating a logarithm by hand? ============

Aim: To find C:=log_{A}(B) where 1<B<A :
one defines two sequences

a_2,a_3,...

n_1,n_2,... (=positive integers) ,

by means of relations

(a_i)^{n_i} < a_{i-1} < a_{i})^{1+n_i}

a_{i+1}:= a_{i-1}/(a_i)^{n_i}

a_0:= A , A_1:=B .
Then

C=1/(n_1+1/(n_2+ 1/n_3+... )) - continuous fraction.

This is Shanks algorithm: see
D.Shanks ,"A logarithm algorithm", MTAC 8,no.46(1954)60--64.  Display posts from previous: All Posts1 Day7 Days2 Weeks1 Month3 Months6 Months1 Year Oldest FirstNewest First
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