Error De Ln
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Logarithmic Error Calculation
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Log Uncertainty
how it works: 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
Logarithmic Error Bars
error for $\ln (x)$ (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 uncertainty logarithm base 10 down vote accepted Simple error 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
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 quantities we need to learn how these errors propagate to an error in the result. how to find log error in physics We assume that the two directly measured quantities are X and Y, with errors X and error propagation examples Y respectively. The measurements X and Y must be independent of each other. The fractional error is the value of the error divided error propagation square root by the value of the quantity: X / X. 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 http://physics.stackexchange.com/questions/95254/the-error-of-the-natural-logarithm 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 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: http://www.upscale.utoronto.ca/PVB/Harrison/ErrorAnalysis/Propagation.html 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? What does it remind you of? (Hint: change the delta's to d's.) Question 9.2. A student measures three lengths a, b and c in cm and a time t in seconds: a = 50 ± 4 b = 20 ± 3 c = 70 ± 3 t = 2.1 ± 0.1 Calculate a + b, a + b + c, a / t, and (a + c) / t. Question 9.3. Calculate (1.23 ± 0.03) + . ( is the irrational number 3.14159265 ) Question 9.4. Calculate (1.23 ± 0.03) ×
från GoogleLogga inDolda fältBöckerbooks.google.sehttps://books.google.se/books/about/Oscillatory_brain_activity_and_its_analy.html?hl=sv&id=5vvsnCJoYDoC&utm_source=gb-gplus-shareOscillatory brain activity and its analysis on the basis of MEG and EEGMitt bibliotekHjälpAvancerad boksökningKöp e-bok – 166,33 krSkaffa ett tryckt exemplar av den här bokenWaxmann VerlagAmazon.co.ukAdlibrisAkademibokandelnBokus.seHitta boken i ett bibliotekAlla försäljare»Oscillatory brain activity and its analysis on the basis of MEG and EEGBernd FeigeWaxmann Verlag 0 Recensionerhttps://books.google.se/books/about/Oscillatory_brain_activity_and_its_analy.html?hl=sv&id=5vvsnCJoYDoC Förhandsvisa den här boken » Så tycker andra-Skriv en recensionVi kunde inte hitta några recensioner.Utvalda sidorTitelsidaInnehållIndexInnehållIntroduction 1 Spectral power analysis 23 Eventrelated spectral changes 49 The PhaseAligned Spec 123 Discussion 143 Bibliography 161 Acknowledgements 186 Upphovsrätt Vanliga ord och fraseramplitude analysis window Arguments average avg_q baseline beforetrig Bereitschaftspotential block brain activity Brain Computer Interface C. M. Gray channel names coherence components condition correlation corresponding cortex data set decomposition default desynchronization dipole effect Electroencephalogr Clin Neurophysiol epoch event-related potentials evoked field experimental format Fourier transform function gamma band hemisphere Hz band idling rhythm input integration Isocontour language latency maps matrix maximum measure modality motor movement-related mu rhythm multiple neuromagnetic neuronal Neurosci noise non-language stimuli normalized spectral power number of channels number of epochs offset onset option oscillations output Pfurtscheller phase points posplot potentials processing reference channel relative gain responses sampling frequency script sensor sensorimotor short integers signal significant single specified spectral power values statistical stimulus class subjects subtract synchronization time/frequency tinnitus tion topography tran