Logarithm Error Propagation
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How To Calculate Uncertainty Of Logarithm
Search Search Go back to previous article Username Password Sign in error propagation log base 10 Sign in Sign in Registration Forgot password Expand/collapse global hierarchy Home Core Analytical Chemistry Quantifying Nature Expand/collapse global error propagation ln location Propagation of Error Last updated 20:33, 14 May 2016 Save as PDF Share Share Share Tweet Share IntroductionDerivation of Exact FormulaDerivation of Arithmetic ExampleCaveats and WarningsDisadvantages of
Logarithmic Error Calculation
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 uncertainties from multiple variables, in order to provide an accurate measurement of uncertainty. Introduction Every measurement has an air of uncertainty
Uncertainty Logarithm Base 10
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 uncertainty of the molar absorptivity. This example will be continued below, after the derivation (see Example Calculation). Derivation of Exact Formula Suppose a certain experiment requires multiple instruments to carry out. These instruments each have different variability in their measurements. The results of
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How To Find Log Error In Physics
is a question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute: Sign up Here's error propagation calculator 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 http://chem.libretexts.org/Core/Analytical_Chemistry/Quantifying_Nature/Significant_Digits/Propagation_of_Error 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 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 http://physics.stackexchange.com/questions/95254/the-error-of-the-natural-logarithm 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 r
the quantity. Uncertainty in logarithms to other bases (such as common logarithm error propagation logs logarithms to base 10, written as log10 or simply log) is this absolute uncertainty adjusted by a factor (divided by 2.3 for common logs). Note, logarithms do not have units.
\[ ln(x \pm \Delta x)=ln(x)\pm \frac{\Delta x}{x}\] \[~~~~~~~~~ln((95 \pm 5)mm)=ln(95~mm)\pm \frac{ 5~mm}{95~mm}\] \[~~~~~~~~~~~~~~~~~~~~~~=4.543 \pm 0.053\]12:33 AM #1 Biochemist Newbie Joined: Aug 2016 From: Germany Posts: 5 Thanks: 0 Taking the binary logarithm - error propagation? Hi, How do I calculate the error (in my case represented by the standard deviation) of a set of data which are converted to their binary logarithm? I have for example 10 numerical values for which I can calculate the standard deviation. After converting these 10 values to their binary logarithm, I can either calculate the new standard deviation based on these new 10 values as I did before or I can use the rules of the Gaussian error propagation for calculating the new error (standard deviation). The formula for the latter is shown in the attachment. I get different results using one or the other method. Which way of calculating the standard deviation is correct (and why)? Attached Images Error propagation for binary logarithm.jpg (3.8 KB, 16 views) September 8th, 2016, 02:23 PM #2 romsek Senior Member Joined: Sep 2015 From: CA Posts: 183 Thanks: 104 to make sure I understand this before looking in detail at it. You have 10 samples from an underlying Gaussian distribution $X_n \sim N(\mu_X, \sigma_X)$ You can estimate $\sigma$ the usual way to come up with the sample variance $\hat{\sigma}_X$ Now let a new random variable $Y \sim \log_2(X)$ You can process these samples to obtain $Y_n = \log_2(X_n)$ and you want to determine how to calculate $\hat{\sigma}_Y$, presumably from the $Y_n$ Is this correct? September 9th, 2016, 01:29 AM #3 Biochemist Newbie Joined: Aug 2016 From: Germany Posts: 5 Thanks: 0 Yes, this is correct. This is what I want to do. Actually, I can think of two ways how to do it but I don't know which is the right one. September 9th, 2016, 03:45 PM #4 romsek Senior Member Joined: Sep 2015 From: CA Posts: 183 Thanks: 104 since the underlying distribution is Gaussian, what do you plan to do for negative values who's binary logarithm is undefined? September 9th, 2016, 04:54 PM #5 romsek Senior Memb