Error Analysis Ln
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a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting error propagation natural log ads with us Physics Questions Tags Users Badges Unanswered Ask Question _ Physics Stack Exchange is error propagation ln a question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute: Sign up Here's how it logarithmic error calculation 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
Log Uncertainty
$\ln (x)$ (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 4,72711444 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 logarithmic error bars 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 right, thx for pointing out, ill add
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 quantities we need uncertainty logarithm base 10 to learn how these errors propagate to an error in the result. We
How To Find Log Error In Physics
assume that the two directly measured quantities are X and Y, with errors X and Y respectively. The measurements X
Compound Error Definition
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 / X. The fractional error multiplied by 100 http://physics.stackexchange.com/questions/95254/the-error-of-the-natural-logarithm 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 error in the result of an addition or subtraction is http://www.upscale.utoronto.ca/PVB/Harrison/ErrorAnalysis/Propagation.html 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 to you? What does it remind you of? (Hint: change the delta's to d's.) Question 9.2. A student measu
it. In doing this it is crucial to understand that all measurements of physical quantities are subject to uncertainties. It is never possible to measure anything exactly. It is good, of http://teacher.nsrl.rochester.edu/phy_labs/AppendixB/AppendixB.html course, to make the error as small as possible but it is always there. And in order to draw valid conclusions the error must be indicated and dealt with properly. Take the measurement of a person's height as an example. Assuming that her height has been determined to be 5' 8", how accurate is our result? Well, the height of a person depends on how straight she stands, error propagation whether she just got up (most people are slightly taller when getting up from a long rest in horizontal position), whether she has her shoes on, and how long her hair is and how it is made up. These inaccuracies could all be called errors of definition. A quantity such as height is not exactly defined without specifying many other circumstances. Even if you could precisely specify the "circumstances," error analysis ln your result would still have an error associated with it. The scale you are using is of limited accuracy; when you read the scale, you may have to estimate a fraction between the marks on the scale, etc. If the result of a measurement is to have meaning it cannot consist of the measured value alone. An indication of how accurate the result is must be included also. Indeed, typically more effort is required to determine the error or uncertainty in a measurement than to perform the measurement itself. Thus, the result of any physical measurement has two essential components: (1) A numerical value (in a specified system of units) giving the best estimate possible of the quantity measured, and (2) the degree of uncertainty associated with this estimated value. For example, a measurement of the width of a table would yield a result such as 95.3 +/- 0.1 cm. Significant Figures The significant figures of a (measured or calculated) quantity are the meaningful digits in it. There are conventions which you should learn and follow for how to express numbers so as to properly indicate their significant figures. Any digit that is not zero is significant. Thus 549 has three significant figu