Propagation Of Error Natural Log
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Uncertainty Logarithm Base 10
us Physics Questions Tags Users Badges Unanswered Ask Question _ Physics Stack Exchange is a question and answer site for active how to find log error in physics researchers, academics and students of physics. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and logarithmic error bars rise to the top The error of the natural logarithm up vote 10 down vote favorite 2 Can anyone explain why the error for $\ln (x)$ (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
Absolute Uncertainty Exponents
share|cite|improve this question edited Jan 25 '14 at 20:01 Chris Mueller 4,72811444 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 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 \appr
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Error Calculation Physics
Expand/collapse global hierarchy Home Core Analytical Chemistry Quantifying Nature Expand/collapse global location Propagation of Error Last updated 20:33, 14 May 2016 Save as PDF Share http://physics.stackexchange.com/questions/95254/the-error-of-the-natural-logarithm Share Share Tweet Share IntroductionDerivation of Exact FormulaDerivation of Arithmetic ExampleCaveats and WarningsDisadvantages of Propagation of Error ApproachTreatment of Covariance TermsReferencesContributors Propagation of Error (or Propagation of Uncertainty) is defined as the effects on a function by a variable's uncertainty. It is a calculus derived statistical calculation designed to combine http://chem.libretexts.org/Core/Analytical_Chemistry/Quantifying_Nature/Significant_Digits/Propagation_of_Error uncertainties from multiple variables, in order to provide an accurate measurement of uncertainty. Introduction Every measurement has an air of uncertainty about it, and not all uncertainties are equal. Therefore, the ability to properly combine uncertainties from different measurements is crucial. Uncertainty in measurement comes about in a variety of ways: instrument variability, different observers, sample differences, time of day, etc. Typically, error is given by the standard deviation (\(\sigma_x\)) of a measurement. Anytime a calculation requires more than one variable to solve, propagation of error is necessary to properly determine the uncertainty. For example, lets say we are using a UV-Vis Spectrophotometer to determine the molar absorptivity of a molecule via Beer's Law: A = ε l c. Since at least two of the variables have an uncertainty based on the equipment used, a propagation of error formula must be applied to measure a more exact unce
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 http://www.upscale.utoronto.ca/PVB/Harrison/ErrorAnalysis/Propagation.html quantities we need to learn how these errors propagate to an error in the result. We assume that the two directly measured quantities are X and Y, with errors X and Y 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 / how to 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 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 propagation of error 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: 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
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