What do you mean by "propagate a few uncertainties?"
I need to propagate a few uncertainties for logarithmic numbers, using the following two formulas:
For:
where:
uncertainty in result
uncertainty in numbers used for calculation
numbers used for calculation
AND
For:
where:
uncertainty in result
result of calculation
uncertainty in numbers used for calculation
My question is, what exactly are the 0.434 and 2.303 values for? I assume they're to counter something that log calculations add?
I'm actually using these formulas for chemistry (although they're aren't chemistry specific). In chemistry, or any other science, you have a margin of error in your results, and so you need to calculate the possible range of errors. These formulas are used to calculate the uncertainties in a value, ie. if my value is 60.0, then a possible range of uncertainties is +/- 0.5. I think the formulas are supposed to be "statistical" formulas?
^ Yes, I believe that is both the upper and lower bound for the error of x
The and notation is just the form of the value. The base 10 notation isn't really needed, it's just for conversion and ease of use of the formula.
So, for forms, you would use
And for forms, you would use
Well notation aside, I was able to see where the numbers came from.
Ignoring sign, write
Use approximation that for small , .
And we can see that .
Similarly, for the second part,
and for u close to 0, .
And .
The result is not an upper bound but rather an approximation of an upper bound.
I may not have written/justified as well as possible but that's the general idea.
They've used calculus to convert from , where is the error, to the approximation using
For the natural logarithm, that is particularly easy: If f(x)= ln(x) then f'(x)= 1/x.
So what they did was convert from "common logarithm", , to the natural logarithm: [tex]ln(x)= \frac{log_{10}(x)}{log_{10}(e)}[/quote] where e is the base of the natural logarithm: e is approximately 2.718... and ln(e)= 1.
In any case, they are converting back and forth between base 10 and base "e".
, approximately and