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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.
Aboute
The constant e is a real and an irrational number that has a value, which is approximately equal to 2.71828, when given up to 6 significant digits. Like π, as proved by Charles Hermite, e is a transcendental number^{1}. A transcendental number is either a non-algebraic complex or real number, which is not a root of any non-zero rational polynomial equation. It is most commonly seen as the base of natural exponential function and at the base of a natural logarithm function.[illustration not visible in this excerpt]
Exponential and Logarithm functions
A natural exponential function is a certain kind of function where e is multiplied x times with itself, which intern can be written as x raised to the power e. Therefore,[illustration not visible in this excerpt], is a natural exponential function. This process of raising powers is called exponentiation. Logarithms are the inverse operation to this process of exponentiation. This means that the logarithm function of a number is a certain kind of function in which the exponent is raised to a certain base as to produce the required number. For example when the logarithm of x to the base b gives y (logbx = y), then, by = x where b > 0 and b Ψ 1.
Pickover, Cliff. "The 15 Most Famous Transcendental Numbers - Cliff Pickover".Sprott.physics.wisc.edu. Web. 2015.
When the base of the logarithm function is e then it is termed as a natural logarithm function. When the base of the logarithm function is 10 then it is termed as common logarithm. As astonishing as it sounds these logarithms with different bases have different uses. The common logarithm is most efficaciously and commonly used in spectroscopy and in various engineering fields, whereas the natural logarithm function is generally used in statistics and economics.
The rationale behind the use of natural logarithm instead of common logarithm is justified in the Andrew Gelman and Jennifer Hill’s book on regression. Under the section of the concept of linear regression of social sciences, it was stated that, “We prefer natural logs (that is, logarithms base e) because, as described above, coefficients on the natural-log scale are directly interpretable as approximate proportional differences: with a coefficient of 0.06, a difference of 1 in x corresponds to an approximate 6% difference in y, and so forth.”^{[2]} This means that exp(x)[illustration not visible in this excerpt]^{[3]} in Taylor series expansion for very small values of x.
So, in the example given in the book there is difference of 0.06 on natural logarithm scale, which corresponds to an approximate multiplication of 1.06 on their original scale. This gives a 6% increase.
However, the Taylor series expansion of common logarithms or rather exponents of 10 is comparatively convoluted, which is given by:
illustration not visible in this excerpt
In this case 2.302585 is the value of the natural logarithm of 10. This goes on to state that a change of 0.01 on a common logarithm scale will approximately correspond to increment of 2.3% on the original scale.
I shall be giving more details about the applications of natural logarithm functions and of e later in this paper.
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^{[4]}. 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 [illustration not visible in this excerpt]^{[5]}
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. Following in 1668, there was development on logarithms where Nicolaus Mercator calculated the expansion of log(1+x) in his Logarithmotechnia. He described log to the base e as natural logarithms. But there was no explicit appearance of e yet again. It was only in 1683 that Jacob Bernoulli through his study
on compound interest, and not logarithms, discovered the number e. It is said that he tried to solve a problem in compound interest by calculating a limit of[illustration not visible in this excerpt], where n tends to infinity and the limit existed between 2 and 3, via binomial theorem. However, he failed to see any connections between logarithm and his results. In 1684 James Gregory made connections between exponents and logarithms. In 1690, e explicitly appeared as the alphabet b. “e” first made its appearance as “e” only in 1731 in a letter when Leonhard Euler wrote to Christian Goldbach. In 1748 in his publication ‘Introducilo in Analysin infinitorum’, he showed his various ideas regarding e. He presented an idea that
illustration not visible in this excerpt
Where[illustration not visible in this excerpt]. And calculated the value of e till 18 decimal places.
Some properties of “e” that make it important and serve as a reason as to why it is considered natural and a base for the natural logarithm function. It is now known to us that e is such a number that makes the area under the rectangular hyperbola from 1 to e equal to 1. It is this property of e that makes it the base of natural logarithm function^{[6]}. There are various other ways that e is similarly proved to be natural and thus an apt base for the natural logarithm (ln) function. I am in this paper shall be exploring another way of doing so. Before I do that I shall discuss some properties of e that make it important and makes it a reason as to why it is considered natural and a base for the natural logarithm function.
The derivative of exis ex itself. This is a unique property of e and no other function can have its derivative as the function itself. Also thought the equation е(1Хл:) + 1 = 0, 5 most important number in mathematics are linked and contains fundamental concepts like addition, multiplication, raising to power and equality. ‘e’ is also liked with calculus; through Euler’s equation, which he deduced by the de moiveres formula, elx = cos(x) + isin(x). This equation also leads to Fourier analysis. There are so many other important properties, but the last property I shall be giving is:
illustration not visible in this excerpt
In recognition of the natural description of the properties of ‘e’, the exponential functions and logarithmic functions are called natural.
For an exponential curve, since there is always going to be a tangent, which is dependent on the base of the function, at every point on the curve, I shall be finding out the base of the exponential function for which the tangent can be equal to 1. By doing so I will obtain a function whose derivative will be equal to the function itself. Such a function can be none other than ‘ax’. However, so as to mathematically find such a base, I shall be considering a function ax. I will take close positive and negative estimates for this base ‘a’, and ultimately prove that the true value of ‘a’ for the desired conditions will be given by the common limit of the monotone increasing and monotone decreasing functions. This natural base of exponential functions is also used as the base for the logarithm functions, thus naming it as the natural logarithm function.
I hope to find the value of e using the slope of exponential functions. It is evident from all the exponential functions that for all bases these functions are convex, this means that if we join any two points on the curve, the line segment joining two such points will always be above the curve. Formally expressing this property of exponential curves:
Let there be a point A with coordinates (xlt aXl) and a point В with coordinates x2, aXz for some exponential curve y = ax. Now, suppose that there is a point C at lies in the interior of the curve in such a way that it lies on the line segment AB that divides AB in ratio p: q, where p and q are positive numbers and (p + q = 1) in such a case the coordinates of C will be[illustration not visible in this excerpt]I took p + q = 1 as, when the section formula is applied the denominator would have p + q, so to simplify the calculations while finding the coordinates of C, p + q is taken as 1. For these coordinates of C there will always lie a point D on the curve, directly below C, that will have coordinates[illustration not visible in this excerpt]. Therefore, for convexity, I can correctly say that [illustration not visible in this excerpt].
Graph 1 depicting the convexity
illustration not visible in this excerpt
The slope of the exponential function has no breaks and is always continuous. I can say this because the differentiation of the function gives the slope^{[7]} and therefore [illustration not visible in this excerpt] In[illustration not visible in this excerpt]^{[8]}
Now it is evident that both the exponential function aX are continuous and have no breaks and \na is a constant; thus they are defined at all points. Using this information, I can deduce that there will be a tangent at every point of the curve. The slope of the tangent depends on the base of the function.^{[9]}
To prove this, Let there be a function[illustration not visible in this excerpt]and ma be the gradiesnt of the graph at x = 0. This means that the derivative of the function at point x = 0 is ma. This means that[illustration not visible in this excerpt]
Now, using the first principle,[illustration not visible in this excerpt]
illustration not visible in this excerpt^{[10]}
Therefore, I can write that
illustration not visible in this excerpt
With this established, I am going to find that what should be the base of the exponential function to make the slope of the tangent at the coordinate (0,1) equal to 1. I am going to take
[...]
^{[1]} Pickover, Cliff. "The 15 Most Famous Transcendental Numbers - Cliff Pickover".Sprott.physics.wisc.edu. Web. 2015.
^{[2]} Gelman, Andrew, and Jennifer Hill. DataAnalysis Using Regression and Multilevel/Hierarchical Models. 1st ed. New York: Cambridge UP, 2007. 60-61. Print.
^{[3]} Cook, John D. "Another Reason Natural Logarithms Are Natural." Data Analysis. John D. Cook, 5 Feb. 2015. Web. 2015. <http://www.johndcook.com/blog/2015/02/05/natural- logarithms-are-natural/>.
^{[4]} O' Conner, J J, and E F Robertson. "The Number E." MacTutor History of Mathematics, Sept. 2001. Web. 2015.
^{[5]} O' Conner, J J, and E F Robertson. "The Number E." MacTutor History of Mathematics, Sept. 2001. Web. 2015.
^{[6]} O' Conner, J J, and E F Robertson. "The Number E." MacTutor History of Mathematics, Sept. 2001. Web. 2015.
^{[7]} Zidar, Owen. "A Primer on Derivatives and Maximization Problems." A Primer on Derivatives and Maximization Problems 1 (n.d.): n. pag.Faculty.chicagobooth. Chicagobooth, 2015. Web. 28 2015.
<http://faculty.chicagobooth.edu/owen.zidar/teaching/Fall%202015/Week0/Week0_Primer_On_
Derivatives_and_Maximization.pdf>.
^{[8]} Kahen. "Show That D/dx (aAx) = AAxln A." Calculus. Stack Exchange Inc, May 2013. Web. 2015. <http://math.stackexchange.com/questions/398826/show-that-d-dx-ax-ax-ln-a>.
^{[9]} "Derivative of Exponential Functions." Derivative of Exponential Functions. Geogebra, Web. 2015. <http://webspace.ship.edu/msrenault/GeoGebraCalculus/derivative_exponential_functions.html
>.
^{[10]} Martin, David. "Differentiation from First Principles." Mathematics for the International Student: Mathematics HL (Core). 3rd ed. Adelaide: Haese Mathematics, 2012. 523. Print.
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