Error Propagation In Logarithms
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How To Calculate Uncertainty Of Logarithm
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Logarithmic Error Calculation
only 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 vote favorite 2 Can anyone explain why the error for $\ln (x)$ (where for $x$ we have
Error Propagation Ln
$x\pm\Delta x$) 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 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 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.
\[ ln(x \pm \Delta x)=ln(x)\pm \frac{\Delta x}{x}\] \[~~~~~~~~~ln((95 \pm 5)mm)=ln(95~mm)\pm \frac{ 5~mm}{95~mm}\] \[~~~~~~~~~~~~~~~~~~~~~~=4.543 \pm 0.053\]constant size. Indeterminate errors have unpredictable size and sign, with equal likelihood of being + or -. RULES FOR ELEMENTARY OPERATIONS (DETERMINATE ERRORS) SUM RULE: When R = A + B then ΔR = ΔA + ΔB https://www.lhup.edu/~dsimanek/scenario/errorman/rules.htm DIFFERENCE RULE: When R = A - B then ΔR = ΔA - ΔB PRODUCT RULE: When R = AB then (ΔR)/R = (ΔA)/A + (ΔB)/B QUOTIENT RULE: When R = A/B then (ΔR)/R = (ΔA)/A + (ΔB)/B POWER RULE: When R = An then (ΔR)/R = n(ΔA)/A or (ΔR) = n An-1(ΔA) Memory clues: When quantities are added (or subtracted) their absolute errors add (or subtract). But when quantities are error propagation multiplied (or divided), their relative fractional errors add (or subtract). These rules will be freely used, when appropriate. We can also collect and tabulate the results for commonly used elementary functions. Note: Where Δt appears, it must be expressed in radians. RULES FOR ELEMENTARY FUNCTIONS (DETERMINATE ERRORS) EQUATION ERROR EQUATION R = sin q ΔR = (dq) cos q R = cos q ΔR = -(dq) sin q R = tan error propagation in q ΔR = (dq) sec2 q R = ex ΔR = (Δx) ex R = e-x ΔR = -(Δx) e-x R = ln(x) ΔR = (Δx)/x Any measures of error may be converted to relative (fractional) form by using the definition of relative error. The fractional error in x is: fx = (ΔR)x)/x where (ΔR)x is the absolute ereror in x. Therefore xfx = (ΔR)x. The rules for indeterminate errors are simpler. RULES FOR ELEMENTARY OPERATIONS (INDETERMINATE ERRORS) SUM OR DIFFERENCE: When R = A + B then ΔR = ΔA + ΔB PRODUCT OR QUOTIENT: When R = AB then (ΔR)/R = (ΔA)/A + (ΔB)/B POWER RULE: When R = An then (ΔR)/R = n(ΔA)/A or (ΔR) = n An-1(ΔA) The indeterminate error rules for elementary functions are the same as those for determinate errors except that the error terms on the right are all positive. Students who are taking calculus will notice that these rules are entirely unnecessary. The determinate error equations may be found by differentiating R, then replading dR, dx, dy, etc. with ΔR, Δx, Δy, etc. This is equivalent to expanding ΔR as a Taylor series, then neglecting all terms of higher order than 1. This is a valid approximation when (ΔR)/R, (Δx)/x, etc. are
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