Error Propagation With Log
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Propagation Of Error Antilog
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Error Propagation Ln
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 $\frac{\Delta x}{x}$? I would very much appreciate a error propagation log base 10 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). 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
with: (1) Functions of several variables. (2) Evaluation of partial derivatives, and the chain rules of differentiation. (3) Manipulation of summations
Error Propagation Rules
in algebraic context. At this mathematical level our presentation can be
Derivative Log
briefer. We can dispense with the tedious explanations and elaborations of previous chapters. 6.2 THE CHAIN RULE error propagation for log function AND DETERMINATE ERRORS If a result R = R(x,y,z) is calculated from a number of data quantities, x, y and z, then the relation: [6-1] ∂R ∂R http://physics.stackexchange.com/questions/95254/the-error-of-the-natural-logarithm ∂R dR = —— 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, the differentials dR, dx, dy and dz may be replaced by the https://www.lhup.edu/~dsimanek/scenario/errorman/calculus.htm 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 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 fthe quantity. Uncertainty in logarithms to other bases (such as common error propagation log 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.
\[ ln(x \pm \Delta x)=ln(x)\pm \frac{\Delta x}{x}\] \[~~~~~~~~~ln((95 \pm 5)mm)=ln(95~mm)\pm \frac{ 5~mm}{95~mm}\] \[~~~~~~~~~~~~~~~~~~~~~~=4.543 \pm 0.053\]Community Forums > Mathematics > General Math > Dismiss Notice Join Physics Forums Today! The friendliest, high quality science and math community on the planet! Everyone who loves science is here! Error propagation log base 2 Jun 6, 2009 #1 dipluso Hi, I am trying to represent my data in log2 form rather than "fold change" and I am embarrassed to say I can't remember how convert the error. For example, I have x=3.96 (mean), deltax=0.28 (standard dev). Thus, log2(x)=0.598. But how do I convert the error?? I know how to do it for log10: 0.434(deltax/x) but I can't for the life of me remember how to derive it for an arbitrary base. Any help/pointers much appreciated. Thanks! Last edited: Jun 6, 2009 dipluso, Jun 6, 2009 Phys.org - latest science and technology news stories on Phys.org •The mathematics of music history •One reason so many scientific studies may be wrong •Game theory research reveals fragility of common resources Jun 6, 2009 #2 dipluso Duh, figured it out. For y=ln(x), error dy=dx/x since lob(x) base b = ln(x)/ln(b) Then for y=lob(x) base b , error is dy=(1/ln(2))(deltax/x) Done. Cheers, -Alex dipluso, Jun 6, 2009 (Want to reply to this thread? Log in or Sign up here!) Show Ignored Content Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook Have something to add? Blaming Government for Teacher and Scientist Failures in Integrity Orbital Precession in the Schwarzschild and Kerr Metrics Digital Camera Buyer’s Guide: Real Cameras Polymer Physics and Genetic Sequencing Spectral Standard Model and String Compactifications 11d Gravity From Just the Torsion Constraint Grandpa Chet’s Entropy Recipe Precession in Special and General Relativity Ohm’s Law Mellow LHC Part 4: Searching for New Particles and Decays Struggles with the Continuum – Conclusion Similar Discussions: Error propagation log base 2 Error propagation (Replies: 1) Error Propagation (Replies: 3) X^2*log(2)x=256^2 (2) refers to base 2Is ther any rule to solve (Replies: 6) Error propagation question (Replies: 0) Log base 2 is the same thing as square root?