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Math Help - approximate log(1-x)

  1. #1
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    approximate log(1-x)

    Can I approximate log(1-x) with -x, if my x \leq 0.5 ?

    If i have an optimization problem: maximize A*x + B*log(1-x)
    can I exchange with a problem: maximize A*x + B*(-x) ?
    Last edited by robustor; December 5th 2011 at 04:43 PM.
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  2. #2
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    Re: approximate log(1-x)

    I don't think you could, have you graphed both functions? This should give you a better understanding.
    Last edited by pickslides; December 5th 2011 at 04:36 PM. Reason: *sp
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  3. #3
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    Re: approximate log(1-x)

    It was a typo. I meant -x.

    I plotted both functions and they look very similar for x<0.5
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    Re: approximate log(1-x)

    Why can't you just find \frac{d}{dx}(\ln(1-x)) as part of the overall derivative?
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    Re: approximate log(1-x)

    \frac{1}{x-1}
    What does that tell me?
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  6. #6
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    Re: approximate log(1-x)

    Quote Originally Posted by robustor View Post
    Can I approximate log(1-x) with -x, if my x \leq 0.5 ?

    If i have an optimization problem: maximize A*x + B*log(1-x)
    can I exchange with a problem: maximize A*x + B*(-x) ?
    I suggest you use a truncated version of the MacLaurin Series \displaystyle \begin{align*} \ln{\left(1 - x\right)} = -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \dots \end{align*} which is convergent when \displaystyle \begin{align*} -1 \leq x < 1 \end{align*} .

    So theoretically, yes you could truncate it after the first term, but it won't be very accurate. I would suggest truncating it after three or four terms.
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    Re: approximate log(1-x)

    For small positive values of x, lets say x<=0.5, or x<=0.25 can I approximate x^2 and x^3 with some function of x?

    The reason why I ask is that I cannot have x to the power of something in my objective function
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  8. #8
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    Re: approximate log(1-x)

    Quote Originally Posted by robustor View Post
    For small positive values of x, lets say x<=0.5, or x<=0.25 can I approximate x^2 and x^3 with some function of x?

    The reason why I ask is that I cannot have x to the power of something in my objective function

    Prove it has given you the series of expansion. For |x| < 1 increasing powers of x will get smaller so you can simply say high powers are roughly 0 compared to low powers and discard the former.
    At which value of x this applies depends on how accurate your function needs to be.
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  9. #9
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    Re: approximate log(1-x)

    Sure, i understand that.

    My values of x are 0<x<0.25

    I can then approximate: \log(1-x)= - x - \frac{x^2}{2}

    My question was, can I approximate it with something like:

    \log(1-x)= - x - f(x) , where f(x) is a linear function of x that is smaller than \frac{x^2}{2}?
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