# 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 with uncertainty , the uncertainty on is:

.

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

For the special case of a function of a single variable , the uncertainty on , , is related to the uncertainty on , , through the formula

In our case, we have . The derivative is then

and our uncertainty propagation formula yields

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!