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#1 2006-01-09 15:18:50

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,566

ln(x) approximation

I came up with this approximation for the ln(x).

1000/2(x^(1/1000) - 1/(x^(1/1000)))

or shown in LaTex as:

On the graph, click on it for bigger, the green dots are the above function,
while the red dots is the ln(x).

View Image: mathisfun26.jpg

igloo myrtilles fourmis

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#2 2006-01-09 17:25:12

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,552

Re: ln(x) approximation

Intriguing ... !

There may be a reason for it. Something to do with e, maybe.

Could you plot the error so we can see? There may be more terms in an expansion.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#3 2006-01-09 20:12:48

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,152

Re: ln(x) approximation

An approximation for ln(x)! Funny!
Logarithms were invented for helping us in calculations by approximation.
MathsIsFun is right, it appears a relation exists between the approximation and e.


Character is who you are when no one is looking.

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#4 2006-01-10 04:25:30

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: ln(x) approximation

That's quite an amazing approximation.  Even at values of 100000, it's only ~0.0007 off.

Try to graph

If you can fit that curve to a known and non-log function, you've got an exact approximation.

However, I believe:

Which would just mean that the curve of the difference goes off to infinity.  So I'm thinking either a polynomial or exponential.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#5 2006-01-11 04:38:51

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,566

Re: ln(x) approximation

Okay, now I'll give away how I came up with this.
I started off with the assumption that x^-.99 is similar to x^-1,
and then integrated.
I noted that the results didn't line up as I expected, so I figured it was the "+C" you get when you do an integral.
Anyway, I used the ln(55) for a test, and found I was off by around a 100.
Then I decided to average two curves.
the integral of x^-1.01 and x^-.99, so I did that.
Then my equation was pretty good, but I changed the 100's to 1000's to make it more accurate.
Which is like starting off with x^-1.001 and x^-.999, probably.
So these curves above and below ln(x) are not exactly equidistant, so that's why the answer is not perfect.


igloo myrtilles fourmis

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