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Formula of a line in log-log coordinate system

Posted on 2004-10-26
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Last Modified: 2008-02-01
I have a line in log-log coordinate system that means all of it's points are known. In linear system I can simply write the formula of a line. Would be it possible in this case too?

Thanks for your help!

wbr
Janos
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Question by:kacor
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13 Comments
 
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Assisted Solution

by:d-glitch
d-glitch earned 1000 total points
ID: 12413091
The eqatuion of the line has to be some thing like

                Y = Ao * X^ n

The slope of the line is n.

Once you know n, you can find Ao by inspection.
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by:
wytcom earned 1000 total points
ID: 12413117
Your log-log graph shows log(y) vs log(x).

Let s = log(y)
Let t = log(x)

For a line in the log-log graph we have
s = m*t + b

where m is the slope and b is the intecept with the vertical axis.

So
log(y) = m*log(x) + b

Or
y = exp( m* log(x) + b )
  = exp(b)*x^m
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by:wytcom
ID: 12413125
My (later) post agrees with d-glitch.  Good fast job there, d-glitch!
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LVL 27

Expert Comment

by:d-glitch
ID: 12414882
Thanks wytcom.  I just wish I could spell  "eqatuion"  under pressure.

>> kacor

     It may not be immediately obvious, but function that generates a straight line on a log/log plot can only have one term.

     One way to see this is to realize that you only have two degrees of freedom (slope and intercept) and they wind up being gain and exponent.

      Another way is to realize that if there are more than one term, you won't get a straight line.
      The highest order term will dominate for large X, and the lowest order term will dominate for small X.

              For example:         Y  =  X^2 + B    has slope of 2 for X>>B, and a slope of 0 for X<<B

     
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Author Comment

by:kacor
ID: 12427724
d-glitch and wytcom,

thanks for your help.

wbr
Janos
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LVL 10

Author Comment

by:kacor
ID: 12438137
wytcom wrote in {http:#12413117}:

log(y) = m*log(x) + b

this is not true I mean because b is not value of a number but a length of log(b) (eg. distance, measured on logaritmic scale)

what do you think about?

Janos
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LVL 27

Expert Comment

by:d-glitch
ID: 12438288
Working forward this time:

               Y  =  Ao * X^m       This is a simple power law.


To plot this on log-log paper you have to start by taking the log of each side:

      log( Y)  =  log( Ao * X^m)    

                 =  log ( Ao)  +  log( X^m)             Use the properties of logs

                 =  log ( Ao)  +  m * log( X)             "         "           "         "

                 =  m * log( X)  +  log ( Ao)            Rearranage to see that this is in the form of  y = m*x + b  ==>  which is a straight line.
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Author Comment

by:kacor
ID: 12438355
that's right but my question is which value have I to use here: b or log(b)
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Author Comment

by:kacor
ID: 12438443
and some more:

If you know 2 points of the line you can calculate the slope:

m = (log(y2)-log(y1))/(log(x2)-log(x1))  
from this comes
m = log (y2/y1)/log(x2/x1))

but I don't know really that this m rests the same or I have to transform it too

and the last question is the question of either b or log(b) because:

if it is given the slope and the point P1(x1;y1) then b is calculable as follows

m*log(x) + b = 0 ->> m=-b/log(x)
or
m*log(x) + log(b) = 0 ->> m=-log(b)/log(x)  

which have I to choose??
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LVL 27

Expert Comment

by:d-glitch
ID: 12440019
If you have two points on a log-log plot, you should draw a line through them
and continue the line through the point  [ log(x)=0, log(Ao)] ==> [ x=1, Ao]

==================================================

Given two points, you can find m:

(log(y2)-log(y1))/(log(x2)-log(x1))   = m
----------------------------------------------------

Then you can find the gain term, since the line must also go through the point [ x=1, Ao]

(log(Ao)-log(y1))/(log(1)-log(x1))    = m
                                                         

log(Ao)  = m*[(log(1)-log(x1)] + log(y1)

             =  m*log(1/x1) + log(y1)


                       y1
     Ao    =   ---------
                    x1^m
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Expert Comment

by:wytcom
ID: 12444202
b = log(y) as read off your log-log graph at the point where the line intersects with the vertical axis
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LVL 7

Expert Comment

by:wytcom
ID: 12444220
This is correct:

m = (log(y2)-log(y1))/(log(x2)-log(x1))  
from this comes
m = log (y2/y1)/log(x2/x1))
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