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In science proving something by making it linear is a common?

  1. Feb 10, 2007 #1
    In science proving something by making it linear is a cmmon approach.

    For instance you can write some laws on this form.

    y=a*log(x)+b

    But there lies my problem, because I cannot see how this should represent a linear line. So please does this represent a line? ANd if yes how come?
     
  2. jcsd
  3. Feb 10, 2007 #2
    That does not represent a linear function of x.
     
  4. Feb 10, 2007 #3

    Gib Z

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    Perhaps a is of the form 1/(n log x)...>.<"
     
  5. Feb 10, 2007 #4

    HallsofIvy

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    If you let X= log(x), then y= aX+ b is certainly linear. I imagine that is what you are thinking of. If you have data that does not appear to fall on a straight line when you plot (x,y), plotting (X, y) with X= log(x) may help. Sometimes plotting (log(x), log(y)) will help.

    Stationery stores used to sell "semi-log graph paper" and "log-log graph paper" that already had one or both axes marked in term of the logarithm so that the conversion was automatic.
     
  6. Feb 10, 2007 #5

    D H

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    Are you thinking of a linear approximation? Scientists and engineers use frequently use a linear approximation to make a problem more amenable to analysis.

    In general, one makes a linear approximation to some function about some central point as the first-order Taylor expansion of the function at that point. For a one-dimensional function [itex]y(x)[/itex], the linear approximation for [itex]x\approx x_0[/itex] is

    [tex]y(x) \approx y(x_0) + (x-x_0)\left.\frac{dy}{dx}\right|_{x=x_0}[/tex]

    For your simple example, [itex]y=a\log x + b[/itex], the linear approximation for [itex]x\approx x_0[/itex] is

    [tex] y(x) \approx a \log x_0 + b + a\frac {x-x_0}{x_0} = \frac a {x_0} x + a(\log x_0-1) + b[/tex]

    Linearizing around [itex]x_0=1[/itex] yields a very simple result:

    [tex] y(x) \approx a (x-1) + b[/tex]
     
    Last edited: Feb 10, 2007
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