Uncertainty for natural logarithms

A formula for propagating uncertainties through a natural logarithm

We have been using the Monte Carlo method to propagate errors thus far, which is one of the most powerful and versatile methods out there. However, in this case, we will have a lot of points that need errors propagated through a natural logarithm (one data point for each day!). It turns out that for this special case, there is a formula for determining uncertaintiy:

For data y with uncertainty \sigma_y, the uncertainty on \ln y is:

\sigma_{\ln y} = \frac{\sigma_y}{y}.

If you are curious where this formula comes from (optional calculus!)

For the special case of a function of a single variable f(z), the uncertainty on f, \sigma_f, is related to the uncertainty on z, \sigma_z , through the formula

\sigma_f = \sqrt{ \left( \frac{df}{dz} \right)^2 \sigma_z^2 }

In our case, we have f = \ln z. The derivative df/dz is then

\frac{df}{dz} \rightarrow \frac{d}{dz} \ln z = \frac{1}{z}

and our uncertainty propagation formula yields

\sigma_f = \sqrt{ \left( \frac{df}{dz} \right)^2 \sigma_z^2 }

\sigma_f = \sqrt{ \left( \frac{1}{z} \right)^2 \sigma_z^2 }

\sigma_f = \sqrt{ \left( \frac{\sigma_z}{z} \right)^2 }

\sigma_f = \frac{\sigma_z}{z}

If you know calculus, you can now find the uncertainty for any function of a single variable! If you have more than one variable (like our nickel volume in lab 1 which had a height and a radius), you CAN NOT just add the uncertainties together: you have to do the Monte Carlo!

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Physics 132 Lab Manual by Brokk Toggerson and Aidan Philbin is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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