I have a set of observational measurements, each with their unique +/- observational error. I need to take the logarithm of this set of data. My question is, to include the error bars in the logarithm graph is it appropriate to take the logarithm of the values along with the logarithm of their respective errors? For example is it appropriate to say X=1000 +/- 10 and log(x)= 3 +/- 1 ?
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2 Answers
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In log-space, you interpret errors in terms of orders of magnitude. So, $3 \pm 1$ as a log-error represents a difference of one order of magnitude from $10^3$, i.e. the interval $[10^2, 10^4]$, which is dramatic, assuming that your data $X = 10^3 \pm 10$.
You should take the $\log$ of min and max values of $X$, i.e. if $X = 10^3 \pm 10$, then $\log(X) \in [2.995635, 3.004321]$.
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Propagation of error (a.k.a. propagation of uncertainty) should be used so that you're plotting the relative error. For Log10, that's:
$$ \frac{\sigma_x}{x \times \ln10} $$
where $\sigma_x$ is the error and $x$ is the measurement.
So your example of $1000 \pm 10$ becomes
$$\log_{10}{1000} \pm \frac{10}{1000 \times \ln10}$$
$$3 \pm 0.00434$$
References:
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$\begingroup$ What are $\sigma_x$ and $x$ above? How do they correspond to X and observation error? $\endgroup$– dimitriyCommented Jul 25, 2023 at 23:48
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