Error Analysis Lnx
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Log Uncertainty
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 $\frac{\Delta x}{x}$? I would very much logarithmic error bars 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). Note that sometimes $\left| \frac{\text{d}f(x)}{\text{d}x}\right|$ is used to avoid gettin
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 to learn how these errors propagate to an error in the result. We assume that the two directly propagation of error trig functions measured quantities are X and Y, with errors X and Y respectively. The measurements X and Y
Error Propagation Sine
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
Error Propagation Cosine
fractional error multiplied by 100 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 http://physics.stackexchange.com/questions/95254/the-error-of-the-natural-logarithm 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 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 http://www.upscale.utoronto.ca/PVB/Harrison/ErrorAnalysis/Propagation.html 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 measures three lengths a, b and c in cm and a time t in seconds: a = 50 ± 4 b = 20 ± 3 c = 70 ± 3 t = 2.1 ± 0.1 Calculate a + b, a + b + c, a / t, and (a + c) / t. Question 9.3. Calculate (1.23 ± 0.03) + . ( is the irrational number 3.14159265 ) Question 9.4. Calculate (1.23 ± 0.03) × . Exercise 9.1. In Exercise 6.1 you measured the thickness of a hardcover book. What is the volume of that book? What is the error in that estimated volume? You may have noticed a useful property of
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