Logarithm Error
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Error Propagation Log Base 10
_ Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Join them; it only uncertainty logarithm base 10 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 rise to the top The error of the natural logarithm up vote 10 down
Logarithmic Error Bars
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 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 error propagation ln 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 $$ \Delta \log_b x \approx \frac{\Delta x}{x \cdot \ln b} $$ 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
with: (1) Functions of several variables. (2) Evaluation of partial derivatives, and the chain rules of differentiation. (3) Manipulation of summations in algebraic context. At this mathematical level our presentation can be briefer. We can dispense with the tedious explanations and elaborations
How To Find Log Error In Physics
of previous chapters. 6.2 THE CHAIN RULE AND DETERMINATE ERRORS If a result R =
Error Propagation Derivative
R(x,y,z) is calculated from a number of data quantities, x, y and z, then the relation: [6-1] ∂R ∂R ∂R dR = —— error propagation rules dx + —— dy + —— dz ∂x ∂y ∂z
holds. This is one of the "chain rules" of calculus. This equation has as many terms as there are variables. Then, if the fractional errors are small, http://physics.stackexchange.com/questions/95254/the-error-of-the-natural-logarithm the differentials dR, dx, dy and dz may be replaced by the absolute errors ΔR, Δx, Δy, and Δz, and written: [6-2] ∂R ∂R ∂R ΔR ≈ —— Δx + —— Δy + —— Δz ∂x ∂y ∂z Strictly this is no longer an equality, but an approximation to DR, since the higher order terms in the Taylor expansion have been neglected. So long as the errors are of the order of a few https://www.lhup.edu/~dsimanek/scenario/errorman/calculus.htm percent or less, this will not matter. This equation is now an error propagation equation. [6-3] Finally, divide equation (6.2) by R: ΔR x ∂R Δx y ∂R Δy z ∂R Δz —— = —————+——— ——+————— R R ∂x x R ∂y y R ∂z z The factors of the form Δx/x, Δy/y, etc are relative (fractional) errors. This equation shows how the errors in the result depend on the errors in the data. Eq. 6.2 and 6.3 are called the standard form error equations. They are also called determinate error equations, because they are strictly valid for determinate errors (not indeterminate errors). [We'll get to indeterminate errors soon.] The coefficients in Eq. 6.3 of the fractional errors are of the form [(x/R)(∂R/dx)]. These play the very important role of "weighting" factors in the various error terms. At this point numeric values of the relative errors could be substituted into this equation, along with the other measured quantities, x, y, z, to calculate ΔR. Notice the character of the standard form error equation. It has one term for each error source, and that error value appears only in that one term. The error due to a variable, say x, is Δx/x, and the size of the term it appears in represents the size of that error's contribution to the error in the result, R.propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based https://en.wikipedia.org/wiki/Propagation_of_uncertainty on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g., instrument precision) which propagate to the combination of variables in the function. The uncertainty u can be expressed in a number of ways. It may be defined by the absolute error Δx. Uncertainties can error propagation also be defined by the relative error (Δx)/x, which is usually written as a percentage. Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, σ, the positive square root of variance, σ2. The value of a quantity and its error are then expressed as an interval x how to calculate ± u. If the statistical probability distribution of the variable is known or can be assumed, it is possible to derive confidence limits to describe the region within which the true value of the variable may be found. For example, the 68% confidence limits for a one-dimensional variable belonging to a normal distribution are ± one standard deviation from the value, that is, there is approximately a 68% probability that the true value lies in the region x ± σ. If the uncertainties are correlated then covariance must be taken into account. Correlation can arise from two different sources. First, the measurement errors may be correlated. Second, when the underlying values are correlated across a population, the uncertainties in the group averages will be correlated.[1] Contents 1 Linear combinations 2 Non-linear combinations 2.1 Simplification 2.2 Example 2.3 Caveats and warnings 2.3.1 Reciprocal 2.3.2 Shifted reciprocal 3 Example formulas 4 Example calculations 4.1 Inverse tangent function 4.2 Re