Error Analysis Natural Log
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Uncertainty Logarithm Base 10
Unanswered Ask Question _ Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute: Sign logarithmic error calculation up Here's 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 error for $\ln (x)$ (where for $x$ we have $x\pm\Delta x$) is simply said to be $\frac{\Delta logarithmic error bars x}{x}$? 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 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
constant size. Indeterminate errors have unpredictable size and sign, with equal likelihood of being + or -. RULES FOR ELEMENTARY OPERATIONS (DETERMINATE ERRORS) SUM RULE: When R = A +
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B then ΔR = ΔA + ΔB DIFFERENCE RULE: When R = error propagation rules A - B then ΔR = ΔA - ΔB PRODUCT RULE: When R = AB then (ΔR)/R = (ΔA)/A +
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(ΔB)/B QUOTIENT RULE: When R = A/B then (ΔR)/R = (ΔA)/A + (ΔB)/B POWER RULE: When R = An then (ΔR)/R = n(ΔA)/A or (ΔR) = n An-1(ΔA) Memory clues: When http://physics.stackexchange.com/questions/95254/the-error-of-the-natural-logarithm quantities are added (or subtracted) their absolute errors add (or subtract). But when quantities are multiplied (or divided), their relative fractional errors add (or subtract). These rules will be freely used, when appropriate. We can also collect and tabulate the results for commonly used elementary functions. Note: Where Δt appears, it must be expressed in radians. RULES FOR ELEMENTARY FUNCTIONS (DETERMINATE ERRORS) EQUATION ERROR https://www.lhup.edu/~dsimanek/scenario/errorman/rules.htm EQUATION R = sin q ΔR = (dq) cos q R = cos q ΔR = -(dq) sin q R = tan q ΔR = (dq) sec2 q R = ex ΔR = (Δx) ex R = e-x ΔR = -(Δx) e-x R = ln(x) ΔR = (Δx)/x Any measures of error may be converted to relative (fractional) form by using the definition of relative error. The fractional error in x is: fx = (ΔR)x)/x where (ΔR)x is the absolute ereror in x. Therefore xfx = (ΔR)x. The rules for indeterminate errors are simpler. RULES FOR ELEMENTARY OPERATIONS (INDETERMINATE ERRORS) SUM OR DIFFERENCE: When R = A + B then ΔR = ΔA + ΔB PRODUCT OR QUOTIENT: When R = AB then (ΔR)/R = (ΔA)/A + (ΔB)/B POWER RULE: When R = An then (ΔR)/R = n(ΔA)/A or (ΔR) = n An-1(ΔA) The indeterminate error rules for elementary functions are the same as those for determinate errors except that the error terms on the right are all positive. Students who are taking calculus will notice that these rules are entirely unnecessary. The determinate error equations may be found
Engineering Medicine Agriculture Photosciences Humanities Periodic Table of the Elements Reference Tables Physical Constants Units and http://chem.libretexts.org/Core/Analytical_Chemistry/Quantifying_Nature/Significant_Digits/Propagation_of_Error Conversions Organic Chemistry Glossary Search site Search Search Go back to previous article Username Password Sign in Sign in Sign in Registration Forgot password 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 Share Share error propagation 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 uncertainties from multiple error analysis natural 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 uncertainty of the molar absorptivity. This ex
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