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Forum index » Science and Technology » Math » Recreational
Logarithm
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Taylor Francis
science forum beginner


Joined: 18 Jul 2006
Posts: 1

PostPosted: Tue Jul 18, 2006 2:43 pm    Post subject: Logarithm Reply with quote

Does anyone know a formula for calculating a logarithm by hand?
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Frederick Williams
science forum addict


Joined: 19 Nov 2005
Posts: 97

PostPosted: Tue Jul 18, 2006 2:50 pm    Post subject: Re: Logarithm Reply with quote

Taylor Francis wrote:
Quote:

Does anyone know a formula for calculating a logarithm by hand?

What base?

--
Remove "antispam" and ".invalid" for e-mail address.
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Lynn Kurtz
science forum Guru


Joined: 02 May 2005
Posts: 603

PostPosted: Tue Jul 18, 2006 7:04 pm    Post subject: Re: Logarithm Reply with quote

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?

Look at

http://hyperphysics.phy-astr.gsu.edu/hbase/math/lnseries.html

--Lynn
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Pontypridd Chess Club (Da
science forum beginner


Joined: 18 Jul 2006
Posts: 1

PostPosted: Tue Jul 18, 2006 8:47 pm    Post subject: Re: Logarithm Reply with quote

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
news:8s6vg.78106$ZW3.59999@dukeread04...
> Does anyone know a formula for calculating a logarithm by hand?
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Mike1161
science forum addict


Joined: 14 Oct 2005
Posts: 50

PostPosted: Wed Jul 19, 2006 12:39 am    Post subject: Re: Logarithm Reply with quote

In article <8s6vg.78106$ZW3.59999@dukeread04>, radiulo@gmail.com says...
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
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alex.lupas@gmail.com
science forum beginner


Joined: 23 Feb 2006
Posts: 47

PostPosted: Thu Jul 20, 2006 5:17 am    Post subject: Re: Logarithm Reply with quote

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.
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