UNVEILING THE POTENTIAL OF THE LOGARITHMIC MEAN: AN INTRODUCTION TO THE DIFFERENCE CALCULUS
Abstract
The logarithmic mean, often referred to as the log-mean, has proven its utility across a wide spectrum of disciplines. In this paper, we explore novel applications and uncover its potential in defining the hyperbolic function. Moreover, we introduce a novel approach, which we term the "difference calculus," for deriving two forms analogous to those produced by differential calculus. Notably, our results using this calculus can yield discrete approximations to those obtained via differential calculus. Our discussions predominantly revolve around economic data, assuming positive and discrete variables unless dealing with differentiability-driven scenarios, where continuity and differentiability are presupposed. Additionally, we primarily employ natural logarithms for simplicity. Consider two positive variables, x0 and x1, representing a base period and a comparison period, respectively. We examine their differences, Δx10 = x1 - x0, and logarithmic differences, Δlogx10 = log(x1/x0) = logx1 - logx0. We denote infinitesimal changes as dx and dlogx, with the assumption of non-zero values for interesting outcomes, even when dealing with finite changes of dependent and independent variables in certain functions.
Keywords:
Logarithmic Mean, Difference Calculus, Hyperbolic Function,, Differential Calculus, Economic DataDownloads
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Copyright (c) 2023 Hiroshi Tanaka
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References
Ang, B. W., & Liu, F. L. (2001). A new energy decomposition method: Perfect in decomposition and consistent in aggregation. Energy, 26, 537–548.
Ang, B. W., & Liu, N. (2007a). Handling zero values in the logarithmic mean Divisia index decomposition approach. Energy Policy, 35, 238–246.
Ang, B. W., & Liu, N. (2007b). Negative-value problems of the logarithmic mean Divisia index decomposition approach. Energy Policy, 35, 739–742.
Balk, B. M. (2002–3). Ideal indices and indicators for two or more factors. Journal of Economics and Social Measurement, 28, 203–217.
Balk, B. M. (2008). Price and quantity index numbers: Models for measuring aggregate change and difference. New York: Cambridge University Press.
Bhatia, R. (2008). The logarithmic mean. Resonance, 13(6), 583–594.
Boole, G. (1958). Calculus of finite differences (4th ed.) edited by J. F. Moulton. New York: Chelsea Publishing Company. (Previous editions published under title: A treatise on the calculus of finite differences. The first edition was appeared in 1860.)
Carlson, B. C. (1972). The logarithmic means. American Mathematical Monthly, 79(6), 615–618.
Diewert, W. E. (2005). Index number theory using differences rather than ratios. American Journal of Economics and Sociology, 64, 311–360.
Dodd, E. L.(1941). Some generalizations of the logarithmic mean and of similar means of two variates which become indeterminate when the two variates are equal. Annals of Mathematical Statistics, 12, 422–428.
Encyclopedia of Mathematics, “Finite-difference calculus, Computational mathematics, Numerical analysis.” [Online] Available: https://www.encyclopediaofmath.org. [Accessed February 13, 2018] [12] Goldberg, S. (1958). Introduction to difference Equations. New York: John Wiley & Sons.
Jia, G., & Cao, J. (2003). A new upper bound of the logarithmic mean. Journal of Inequalities in Pure and Applied Mathematics, 4(4), 1–4.
Jordan, C. (1965). Calculus of finite differences (3rd ed.). New York: Chelsea Publishing Company.
Paterson, W. R. (1984). A replacement for the logarithmic mean. Chemical Engineering Science, 39(11), 1635–1636.
Pittenger, A. O. (1985). The logarithmic mean in n variables. American Mathematical Monthly, 92(2), 99–104.
Spiegel, M. R. (1971). Theory and problems of calculus of finite differences and difference equations. New York: McGraw-Hill.
Stolarsky, K. B. (1975). Generalizations of the logarithmic mean. Mathematics Magazine, 48(2), 87–92.
Sugano, H., & Tsuchida, S. (2002). A new approach to discrete data (in Japanese). Bulletin, Faculty of Business Information Sciences, Jobu University, 1, 29–55.
Tsuchida, S. (1997). A family of almost ideal log-change index numbers. Japanese Economic Review, 48, 324–342.
Tsuchida, S. (2014). Interconvertible rules between an aggregative index and a log-change index. Electronic Journal of Applied Statistical Analysis, 7 (2), 394–415. [Online] Available: http://siba-ese.unisalento.it/index.php/ejasa/issue/view/1259.
