When the concept of logarithms was first introduced to me, a plethora of questions revolved around my mind. My inquisitiveness compelled me to think and ask questions as to where are the practical applications of logarithms, why do we take different bases of these functions and what is the need for natural logarithms. Amongst these questions, one particularly intrigued me: why is e particularly the base of the natural logarithm. Why out of all numbers that exist did we choose e as the base of the natural logarithm function?
I was fascinated by why taking the base e made the normal logarithm a natural logarithm. Therefore, to quench the curiosity of many others like me, I will show through this paper that why e is the correct choice for the base of exponential and natural logarithm functions. I shall also be exploring the most important property of e, via this paper.
Table of Contents
1. Introduction
2. Theory
2.1 About e
2.2 Exponential and Logarithm functions
2.3 History of e
3. Overview
4. Proof
5. Conclusion
Objectives and Topics
The primary objective of this paper is to explore and mathematically demonstrate why the constant 'e' (Euler's number) serves as the most appropriate and "natural" base for exponential and logarithmic functions, while also examining its fundamental properties and historical development.
- Historical evolution of Euler's number 'e'
- Properties of natural exponential and logarithm functions
- Mathematical proof regarding the slope of exponential functions
- The relationship between convexity and the base of exponential curves
- Justification for 'e' as the natural base through limit convergence
Excerpt from the Book
History of e:
The first reference to e was in 1618 by John Napier in a table of an appendix for his work on logarithms. William Oughtred, as many believe, wrote this table. In 1647 the area under a rectangular hyperbola was a found out by Saint Vincent. However many believe that he was unaware of the connection of this with logarithms. It was in 1661 that Christiaan Huygens found out the relationship between logarithms and the rectangular hyperbola (y×x = 1). He also defined a new curve, which he called as logarithmic, but it actually was an exponential curve, as we know today. He also calculated the value of log of e as taken to the base 10, up to 17 decimal places, but it appeared to be a calculation of a certain constant, other than e. So up to 1661, although the number e was used allusively and seen in certain papers and theories it was not explicitly defined or calculated.
Summary of Chapters
Introduction: The author outlines the motivation for the study, focusing on the fundamental curiosity regarding why 'e' is chosen as the base for natural logarithms.
Theory: This section defines 'e' as an irrational, transcendental number and introduces the basic concepts of exponential and logarithmic functions, including a brief historical overview of the discovery of 'e'.
Overview: The author provides a strategic plan for the proof, aiming to identify the base 'a' for an exponential function where the tangent slope equals 1 at a specific point.
Proof: This core chapter utilizes mathematical estimates, convexity properties, and limit theory to demonstrate that the base 'a' converges to 'e' as the domain increases.
Conclusion: The author summarizes the findings, confirming that the unique properties of 'e'—specifically its role in making the tangent slope of an exponential function equal to 1—justify its status as the natural base.
Keywords
Euler's number, e, natural logarithm, exponential function, transcendental number, calculus, derivative, convexity, tangent, base, limit, natural exponentiation, history of mathematics, Leonhard Euler, growth rate
Frequently Asked Questions
What is the core focus of this research paper?
The paper investigates the mathematical and logical reasoning behind why the constant 'e' is designated as the base for the natural logarithm function.
Which mathematical fields are primarily involved in this analysis?
The study relies heavily on calculus, specifically differentiation, limit theory, and the analysis of exponential and logarithmic growth patterns.
What is the primary research goal?
The goal is to prove that 'e' is the unique base for an exponential function that yields a tangent slope of 1 at the coordinate (0,1).
What methodology does the author employ?
The author uses a proof-based approach, utilizing the properties of curve convexity and Taylor series expansions to establish the limit convergence of the base 'a' toward 'e'.
How is the main body of the work structured?
The main body moves from theoretical definitions and historical context to a rigorous mathematical proof, followed by a summary of findings.
Which key terms define this work?
The work is defined by terms such as transcendental number, natural logarithm, exponential function, and limit convergence.
Why is the common logarithm (base 10) mentioned in the study?
The common logarithm is used as a point of contrast to highlight why the natural logarithm is more efficient for mathematical derivations and proportional differences.
How does the author define 'e' historically?
The author traces the history of 'e' from John Napier's work in 1618 through the contributions of Bernoulli and finally to Leonhard Euler's explicit naming of the constant.
- Citar trabajo
- Sumaanyu Maheshwari (Autor), 2016, Euler’s number. Why is Eule's number "e" the basis of natural logarithm functions, Múnich, GRIN Verlag, https://www.grin.com/document/344988
