Logaritma Fonksiyonunun İrrasyonel Fonksiyon İle Temsili ve Bu Temsile Dayalı Üstel Fonksiyon Elde Edilmesi Üzerine

Abstract

Bu çalışmada matematikte çok yaygın bir kullanım alanına sahip olan en temel fonksiyonlardan olan logaritma fonksiyonunun irrasyonel bir fonksiyon ile temsil edilmesi üzerine çalışılmıştır. Bu amaçla logaritma fonksiyonu yerine kullanılabilecek irrasyonel bir fonksiyon önerisi yapılmış, uygun matematik ve nümerik analiz tanım ve yöntemleri çerçevesinde önerilen fonksiyon elde edilmiştir. Elde edilen irrasyonel fonksiyon üzerinden ters fonksiyonu oluşturulmak suretiyle üstel fonksiyonlar için yeni bir yaklaşım elde edilmiştir. Sayısal sonuçlar ve grafikler elde edilerek gerekli değerlendirmeler yapılmıştır.

Keywords

• Logaritma
• Doğal Logaritma
• Napier
• Euler

On the Representation of the Logarithm Function by the Irrational Function and the Obtaining of an Exponential Function Based on This Representation

Abstract

In this study, the representation of the logarithm function, which is one of the most basic functions that has a very common usage area in mathematics, has been studied with an irrational function. For this purpose, an irrational function that can be used instead of a logarithm function has been proposed, and the proposed function has been obtained within the framework of appropriate mathematical and numerical analysis definitions and methods. A new approach has been obtained for exponential functions by constructing the inverse function on the obtained irrational function. Necessary evaluations were made by obtaining numerical results and graphics.

Keywords

• Logarithm
• Natural Logarithm
• Napier
• Euler

References

1. Andrews G. E., Askey R., Roy R. (1999) Special Functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, 1999.
2. Boyer, C. B. and Merzbach, U. C. (1991). "Invention of Logarithms." A History of Mathematics, 2nd ed. New York: Wiley, pp. 312-313.
3. Gautschi, Walter. (2008). On Euler’s attempt to compute logarithms by interpolation: A commentary to his letter of February 16, 1734 to Daniel Bernoulli, Journal of Computational and Appllied Mathematics, 219, no.2, 408-415.
4. Havil, J. (2003). "The Baron's Wonderful Canon." 1.2 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 4-11.
5. Kathleen M. Clark., Clemency Montelle. (2011). "Logarithms: The Early History of a Familiar Function - John Napier Introduces Logarithms," Convergence.
6. Kreyszig E. (1993). Advanced Engineering Mathematics, 7th Edition.
7. Matala-Aho Tapani, Vaananen Keijo, Zudilin Wadim. (2005). New Irrationality Measures for q-Logarithms., Mathematics of Computation, Volume 75, Number 254, P.879-889.
8. Mathews John H. (1992). Numerical Methods for Mathematics, Science and Engineering, 2nd Ed., Prentice-Hall, Inc.

9. Koelink, E., Assche, W.V. (2009). Leonhard Euler and a q-analogue of the logarithm. , American Mathematical Society, V.137, N.5, P.1663-1676.
10. Nofal, Christopher Paul. (2006) Proof that the Natural Logarithm Can Be Represented by the Gaussian Hypergeometric Function.
11. Rice B., Gonzales-Velasco E., Corrigan A. (2017). John Napier. In The Life and Works of John Napier, Springer.
12. Thomas George B.. D. Weir Maurice, Hass Joel R. (2010). Thomas’ Calculus, 12th Edition, Pearson.