Error Propagation Through Logarithm
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Error Propagation For Log Function
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How To Calculate Uncertainty Of Logarithm
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 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 natural log error propagation 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 $$ \frac{\text{d}\ln(x)}{\text{d}x} = \frac{1}{x} $$ 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 answe
the quantity. Uncertainty in logarithms to other bases (such as common http://phys114115lab.capuphysics.ca/App%20A%20-%20uncertainties/appA%20propLogs.htm 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.
\[ ln(x \pm \Delta x)=ln(x)\pm \frac{\Delta x}{x}\] \[~~~~~~~~~ln((95 \pm 5)mm)=ln(95~mm)\pm \frac{ 5~mm}{95~mm}\] \[~~~~~~~~~~~~~~~~~~~~~~=4.543 \pm 0.053\]uncertainty of an answer obtained from a calculation. Every time data are measured, there is an uncertainty associated with that measurement. (Refer to guide to Measurement and Uncertainty.) If these measurements used in your calculation have some uncertainty associated with them, then the final http://physics.appstate.edu/undergraduate-programs/laboratory/resources/error-propagation answer will, of course, have some level of uncertainty. For instance, in lab you might measure an object's position at different times in order to find the object's average velocity. Since both distance and time measurements have uncertainties associated with them, those uncertainties follow the numbers throughout the calculations and eventually affect your final answer for the velocity of that object. How would you determine the uncertainty in your calculated values? In lab, graphs are often used where error propagation LoggerPro software calculates uncertainties in slope and intercept values for you. In other classes, like chemistry, there are particular ways to calculate uncertainties. In fact, since uncertainty calculations are based on statistics, there are as many different ways to determine uncertainties as there are statistical methods. The error propagation methods presented in this guide are a set of general rules that will be consistently used for all levels of physics classes in this department. In the following examples: error propagation for q is the result of a mathematical operation δ is the uncertainty associated with a measurement. For example, if you have a measurement that looks like this: m = 20.4 kg ±0.2 kg Thenq = 20.4 kg and δm = 0.2 kg First Step: Make sure that your units are consistent Make sure that you are using SI units and that they are consistent. If you are converting between unit systems, then you are probably multiplying your value by a constant. Please see the following rule on how to use constants. Logger Pro If you are using a curve fit generated by Logger Pro, please use the uncertainty associated with the parameters that Logger Pro give you. In the above linear fit, m = 0.9000 andδm = 0.05774. The uncertainty should be rounded to 0.06, which means that the slope must be rounded to the hundredths place as well: m = 0.90± 0.06 If the above values have units, don't forget to include them. Constants If an expression contains a constant, B, such that q =Bx, then: You can see the the constant B only enters the equation in that it is used to determine q. Example: F = mg = (20.4 kg)(-9.80 m/s2) = -199.92 kgm/s2 δF/F = δm/m δF/(-199.92 kgm/s2) = (0.2 kg)/(20.4 kg) δF = ±1.96 kgm/s2 δF = ±2 kgm/s2 F = -199.92 kgm/s2 ±1.96kgm/s2 With the answer r
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