Natural Log Error 2
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Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 entries Last updated: Wed Oct 19 2016 Created, developed, and error propagation natural log nurturedbyEricWeisstein at WolframResearch Number Theory>Constants>Miscellaneous Constants> Calculus and Analysis>Special Functions>Logarithms> Discrete Mathematics>Experimental how to calculate uncertainty of logarithm Mathematics> More... Calculus and Analysis>Series>BBP Formulas> MathWorld Contributors>Cloitre> MathWorld Contributors>Plouffe> Less... Natural Logarithm of 2 The natural logarithm error propagation log base 10 of 2 is a transcendental quantity that arises often in decay problems, especially when half-lives are being converted to decay constants. has numerical value (1) (OEIS A002162). The irrationality measure error propagation ln of is known to be less than 3.8913998 (Rukhadze 1987, Hata 1990). It is not known if is normal (Bailey and Crandall 2002). The alternating series and BBP-type formula (2) converges to the natural logarithm of 2, where is the Dirichlet eta function. This identity follows immediately from setting in the Mercator series, yielding (3) It is also a special
Logarithmic Error Calculation
case of the identity (4) where is the Lerch transcendent. This is the simplest in an infinite class of such identities, the first few of which are (5) (6) (E.W.Weisstein, Oct.7, 2007). There are many other classes of BBP-type formulas for , including (7) (8) (9) (10) (11) Plouffe (2006) found the beautiful sum (12) A rapidly converging Zeilberger-type sum due to A.Lupas is given by (13) (Lupas 2000; typos corrected). The following integral is given in terms of , (14) The plot above shows the result of truncating the series for after terms. Taking the partial series gives the analytic result (15) (16) where is the digamma function and is a harmonic number. Rather amazingly, expanding about infinity gives the series (17) (Borwein and Bailey 2002, p.50), where is a tangent number. This means that truncating the series for at half a large power of 10 can give a decimal expansion for whose decimal digits are largely correct, but where wrong digits occur with precise regularity. For example, taking gives a decimal value equal to the secon
0.693\,147\,180\,56} as shown in the first line of the table below. The logarithm in other bases is obtained with the formula log b ( 2 )
Uncertainty Logarithm Base 10
= ln ( 2 ) ln ( b ) . {\displaystyle logarithmic error bars \log _{b}(2)={\frac {\ln(2)}{\ln(b)}}.} The common logarithm in particular is ( A007524) log 10 ( 2 ) ≈ 0.301 029 995 663 how to find log error in physics 981 195. {\displaystyle \log _{10}(2)\approx 0.301\,029\,995\,663\,981\,195.} The inverse of this number is the binary logarithm of 10: log 2 ( 10 ) = 1 log 10 ( 2 ) ≈ 3.321 http://mathworld.wolfram.com/NaturalLogarithmof2.html 928 095 {\displaystyle \log _{2}(10)={\frac {1}{\log _{10}(2)}}\approx 3.321\,928\,095} ( A020862). number approximate natural logarithm OEIS 2 6999693147180559945♠0.693147180559945309417232121458 A002162 3 7000109861228866810♠1.09861228866810969139524523692 A002391 4 7000138629436111989♠1.38629436111989061883446424292 A016627 5 7000160943791243410♠1.60943791243410037460075933323 A016628 6 7000179175946922805♠1.79175946922805500081247735838 A016629 7 7000194591014905531♠1.94591014905531330510535274344 A016630 8 7000207944154167983♠2.07944154167983592825169636437 A016631 9 7000219722457733621♠2.19722457733621938279049047384 A016632 10 7000230258509299404♠2.30258509299404568401799145468 A002392 By Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is https://en.wikipedia.org/wiki/Natural_logarithm_of_2 a transcendental number. Contents 1 Series representations 2 Representation as integrals 3 Other representations 4 Bootstrapping other logarithms 4.1 Example 5 References 6 External links Series representations[edit] ∑ n = 1 ∞ ( − 1 ) n + 1 n = ∑ n = 0 ∞ 1 ( 2 n + 1 ) ( 2 n + 2 ) = ln 2. {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)(2n+2)}}=\ln 2.} ∑ n = 0 ∞ ( − 1 ) n ( n + 1 ) ( n + 2 ) = 2 ln ( 2 ) − 1. {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(n+1)(n+2)}}=2\ln(2)-1.} ∑ n = 1 ∞ 1 n ( 4 n 2 − 1 ) = 2 ln ( 2 ) − 1. {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n(4n^{2}-1)}}=2\ln(2)-1.} ∑ n = 1 ∞ ( − 1 ) n n ( 4 n 2 − 1 ) = ln ( 2 ) − 1. {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n(4n^{2}-1)}}=\ln(2)-1.} ∑ n = 1 ∞ ( − 1 ) n n ( 9 n 2 − 1 ) = 2 ln ( 2 ) − 3 2 . {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n(9n^{2}-1)}}=2\ln(2)-{\tfrac
x); double log (double x); float log (float x); long double log (long double x); double log http://www.cplusplus.com/reference/cmath/log/ (T x); // additional overloads for integral types Compute natural logarithm Returns http://people.duke.edu/~rnau/411log.htm the natural logarithm of x. The natural logarithm is the base-e logarithm: the inverse of the natural exponential function (exp). For common (base-10) logarithms, see log10. Header
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The logarithm transformation Introduction to logarithms Change in natural log ≈ percentage change Linearization of exponential growth and inflation Trend measured in natural-log units ≈ percentage growth Errors measured in natural-log units ≈ percentage errors Coefficients in log-log regressions ≈ proportional percentage changes Introduction to logarithms: Logarithms are one of the most important mathematical tools in the toolkit of statistical modeling, so you need to be very familiar with their properties and uses. A logarithm function is defined with respect to a "base", which is a positive number: if b denotes the base number, then the base-b logarithm of X is, by definition, the number Y such that bY = X. For example, the base-2 logarithm of 8 is equal to 3, because 23 = 8, and the base-10 logarithm of 100 is 2, because 102 = 100. There are three kinds of logarithms in standard use: the base-2 logarithm (predominantly used in computer science and music theory), the base-10 logarithm (predominantly used in engineering), and the natural logarithm (predominantly used in mathematics and physics and in economics and business). In the natural log function, the base number is the transcendental number "e" whose deciminal expansion is 2.718282…, so the natural log function and the exponential function (ex) are inverses of each other. The only differences between these three logarithm functions are multiplicative scaling factors, so logically they are equivalent for purposes of modeling, but the choice of base is important for reasons of convenience and convention, according to the setting. In standard mathematical notation, and in Excel and most other analytic software, the expression LN(X) is the natural log of X, and EXP(X) is the exponential function of X, so EXP(LN(X)) = X and LN(EXP(X)) = X. This means that the EXP function can be used to convert natural-logged forecasts (and their respective lower and upper confidence limits) back into real units. You cannot use the EXP function to directly unlog the error statistics of a model fitted to natural-logged data. You need to first convert the forecasts back into real units and then recalculate the errors and error statistics in real units, if it is important to have those numbers. However, the error statistics of a model fitted to natural-logged data can often be interpreted as approximate measures of percentage error, as explained below, and in situations where logging is appropriate in the first place, it is often of intere