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SEM involving a logarithmic

  1. Feb 16, 2016 #1
    Hi, this isn't exactly a homework question, but this seemed like the most appropriate place to put it.
    1. The problem statement, all variables and given/known data
    I have an equation in the form:
    log(a)=log(b)+c.
    I also have standard errors (SEMs) for b and c. I want to find the standard error for log(a) (i.e. log(a) +/- E(log(a)))

    2. Relevant equations
    I know the SEM of some quantity x, where x:=y+z, is given by Ex=sqrt(Ey^2+Ez^2)

    3. The attempt at a solution
    The problem is really trying to find the error of the log.

    In high school I would have solved it by choseing the largest of:
    abs(log(b+Eb)-log(b)) and abs(log(b-Eb)-log(b)), where Eb is the SEM of b.
    if we let this be g then
    E(log(a))=sqrt(g^2+Ec^2)

    However, given I am doing uni research I am not sure whether this would be acceptable.

    I have also considered making a Monte-Carlo simulation of the problem. Drawing random numbers from the distributions b~N(b,Eb) and c~N(c,Ec) and finding the mean and standard deviation of the simulation. However I would like to get an analytical solution.


    Thanks in advance if anyone can help me out
     
  2. jcsd
  3. Feb 16, 2016 #2

    gneill

    User Avatar

    Staff: Mentor

    When you have a function of several variables with errors a common approach is to take partial derivatives of the function with respect to each variable and then combine the errors in quadrature.

    Say you have ##f(a,b,c)## with ##a ± Δa##, ##b ± Δb##, and ##c ± Δc##. Then the total error is given by:

    $$Δf = \sqrt{\left(\frac{\partial f}{\partial a}\right)^2 Δa^2 + \left(\frac{\partial f}{\partial b}\right)^2 Δb^2 + \left(\frac{\partial f}{\partial c}\right)^2 Δc^2 }$$

    Do you remember how to differentiate the log function?
     
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