Error Propagation Of Lnx
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Logarithmic Error Calculation
question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute: Sign up Here's how it works: error propagation log base 10 Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top The error of the natural logarithm up vote 10 down vote favorite 2 Can anyone explain why the error for $\ln error propagation example problems (x)$ (where for $x$ we have $x\pm\Delta x$) is simply said to be $\frac{\Delta x}{x}$? I would very much appreciate a somewhat rigorous rationalization of this step. Additionally, is this the case for other logarithms (e.g. $\log_2(x)$), or how would that be done? error-analysis share|cite|improve this question edited Jan 25 '14 at 20:01 Chris Mueller 4,72711444 asked Jan 25 '14 at 18:31 Just_a_fool 3341413 add a comment| 2 Answers 2 active oldest votes up vote 17 down vote accepted Simple error
Error Propagation Sine
analysis assumes that the error of a function $\Delta f(x)$ by a given error $\Delta x$ of the input argument is approximately $$ \Delta f(x) \approx \frac{\text{d}f(x)}{\text{d}x}\cdot\Delta x $$ The mathematical reasoning behind this is the Taylor series and the character of $\frac{\text{d}f(x)}{\text{d}x}$ describing how the function $f(x)$ changes when its input argument changes a little bit. In fact this assumption makes only sense if $\Delta x \ll x$ (see Emilio Pisanty's answer for details on this) and if your function isnt too nonlinear at the specific point (in which case the presentation of a result in the form $f(x) \pm \Delta f(x)$ wouldnt make sense anyway). Note that sometimes $\left| \frac{\text{d}f(x)}{\text{d}x}\right|$ is used to avoid getting negative erros. Since $$ \frac{\text{d}\ln(x)}{\text{d}x} = \frac{1}{x} $$ the error would be $$ \Delta \ln(x) \approx \frac{\Delta x}{x} $$ For arbitraty logarithms we can use the change of the logarithm base: $$ \log_b x = \frac{\ln x}{\ln b}\\ (\ln x = \log_\text{e} x) $$ to obtain $$ \Delta \log_b x \approx \frac{\Delta x}{x \cdot \ln b} $$ share|cite|improve this answer edited Jan 26 '14 at 7:54 answered Jan 25 '14 at 18:39 LeFitz 48949 1 This (nice) answer is correct for the case that $\Delta x\ll x$ but will otherwise fail; see my answer below for why and what to do there. –Emilio Pisanty Jan 25 '14 at 21:29 very right, thx for pointing out, ill add a short note to
to get a speed, or adding two lengths to get a total length. Now that we have learned how to determine the error in the directly measured log uncertainty quantities we need to learn how these errors propagate to an error in error propagation cosine the result. We assume that the two directly measured quantities are X and Y, with errors X and Y
Logarithmic Error Bars
respectively. The measurements X and Y must be independent of each other. The fractional error is the value of the error divided by the value of the quantity: X / X. http://physics.stackexchange.com/questions/95254/the-error-of-the-natural-logarithm The fractional error multiplied by 100 is the percentage error. Everything is this section assumes that the error is "small" compared to the value itself, i.e. that the fractional error is much less than one. For many situations, we can find the error in the result Z using three simple rules: Rule 1 If: or: then: In words, this says that the error in http://www.upscale.utoronto.ca/PVB/Harrison/ErrorAnalysis/Propagation.html the result of an addition or subtraction is the square root of the sum of the squares of the errors in the quantities being added or subtracted. This mathematical procedure, also used in Pythagoras' theorem about right triangles, is called quadrature. Rule 2 If: or: then: In this case also the errors are combined in quadrature, but this time it is the fractional errors, i.e. the error in the quantity divided by the value of the quantity, that are combined. Sometimes the fractional error is called the relative error. The above form emphasises the similarity with Rule 1. However, in order to calculate the value of Z you would use the following form: Rule 3 If: then: or equivalently: For the square of a quantity, X2, you might reason that this is just X times X and use Rule 2. This is wrong because Rules 1 and 2 are only for when the two quantities being combined, X and Y, are independent of each other. Here there is only one measurement of one quantity. Question 9.1. Does the first form of Rule 3 look familiar to you? Wh
Community Forums > Science Education > Homework and Coursework Questions > Introductory Physics Homework > Not finding help here? Sign up for a free 30min tutor trial with https://www.physicsforums.com/threads/errror-uncertainty-for-ln-x.725440/ Chegg Tutors Dismiss Notice Dismiss Notice Join Physics Forums Today! The friendliest, https://www.chegg.com/homework-help/questions-and-answers/compute-error-propagation-formula-ln-t-ln-l-t-2-pi-l-g-1-2-b-least-squares-y-ln-t-x-ln-x-o-q13397465 high quality science and math community on the planet! Everyone who loves science is here! Errror uncertainty for ln(x) Nov 28, 2013 #1 johnnnnyyy 1. The problem statement, all variables and given/known data What would be the error uncertainty when you take ln of a number. For example ln(10) and the error propagation error uncertainty for 10 is ± 1 2. Relevant equations 3. The attempt at a solution Is the error uncertainty just (1/10)*2.3? (2.3 is the answer to ln(10)) johnnnnyyy, Nov 28, 2013 Phys.org - latest science and technology news stories on Phys.org •Game over? Computer beats human champ in ancient Chinese game •Simplifying solar cells with a new mix of materials •Imaged 'jets' error propagation of reveal cerium's post-shock inner strength Nov 28, 2013 #2 rude man Homework Helper Insights Author Gold Member d/dx ln x = 1/x what is d(ln x) in terms of dx? That's for small changes in x. For larger changes in x, say x → x+a, you can Taylor-series-expand ln (x + a). The first term is of course the above = ln x + a/x , then the higher terms give you better accuracy. In your case, x = 10 and a = 1, the small-change approximation above gets you to within 0.9980 of the correct answer. Adding just 1 extra term in the series gets you to within 0.9999 of the exact answer. Etc. rude man, Nov 28, 2013 Nov 28, 2013 #3 NascentOxygen Staff: Mentor johnnnnyyy said: ↑ 1. The problem statement, all variables and given/known data What would be the error uncertainty when you take ln of a number. For example ln(10) and the error uncertainty for 10 is ± 1 So the correct answer lies between ln(9) and ln(11)? NascentOxygen, Nov 28, 2013 Nov 28, 2013 #4 rude man Homework Helper Insights Auth
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