# Trivial question on differentiating logarithms

#### JC2000

In error analysis however, we work with squares and the sign disappears.
No, they don't. In post #2, BvU misunderstood what you were asking about.
So which is it now I wonder...

#### wrobel

A cruel offtop was here. I should have read initial topic carefully. Sorry

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#### BvU

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We have a few issues happily mixing up to a very confusing thread • differentiation
• propagation of errors linearly
• propagation of errors in quadrature
•  and now wrobel on logatrithms ... oh boy Mark is into differentiation and makes no mistakes (except perhaps slightly overlooking the error analysis context as hinted at in #2 and confirmed in #3)

Your book adds errors linearly and vela and I are in the quadrature camp - so we slightly missed noticing the different approach in your book (as obvious from #14).

My motto is 'try a simple example'.

To avoid sign issues I multiply $A \pm \Delta A$ and $B \pm \Delta B$. Result $AB$.

Worst case down $(A - \Delta A) (B- \Delta B) = AB - B\Delta A- A\Delta B$ to first order in $\Delta$ (i.e. assuming $\Delta A \Delta B <<$ other terms).

Worst case up $(A + \Delta A) (B+ \Delta B) = AB + B\Delta A+ A\Delta B$.

So worst case ${\Delta AB\over AB} = {\Delta A\over A} + {\Delta B\over B}$ -- as in your book I hope.

Check this two ways:
1. in excel multiply $100 \pm 5$ and $120 \pm 12$ or something in Excel or on a calculator.
2. on a piece of paper draw a rectangle of 100 x 120 and the lines at 95, 105 and 108, 132
Worst case is pessimistic: if the errors are independent the probability that both are way off is smaller: the probability distributions have to be combined and then the squares come in.

You can imagine the probability distribution around the 100 as a vertical shading in gray-ish with the darkest grey at 100 and a bit lighter further away. Idem horizontally at 120.
The probability distribution of the product then becomes a gaussian hat around 100x120.

Hmm, googling some picturesmight be useful here. No time now. Work Lets first synchronize at this point -- all clear and all agreed ? • JC2000

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