Why E Is The Base Of The Natural Logarithm1-1

Surname 1

Name

Professor

Title

Date

Why e is the base of the natural logarithms

Introduction

The number e is a constant used as the base of the natural logarithms. The number is

approximately 2.71828. The number is equal to the limit of the  









when the value of n

approaches the infinity. The value is also found from the following expression.













The number is unique in various ways. If the graph with the function 



is plotted, the value

of the slope of the graph at the point x= 0 is equal to the constant number 2.71828. Most of the

calculus functions use the natural logarithmic functions with the base e. The logarithmic function

is the inverse function if the natural exponential expression is used. The natural logarithm of the

integers is calculated with the use of the area under the curve technique if the limits of the

integration are  and the   where k is the positive integer.

A standout amongst the most central constants in science is the Exponential Constant, e, which is

now and again referred to as the Euler's number. The steady, e, is likewise utilized as the base of

natural logarithms. The most intriguing part of the exponential function is its normal

development property. The exponential capacity is said to be the base of numerous development

designs. Nonetheless, the growth property in the exponential constant has not gotten much

spotlight, and numerous perspectives stay uncertain.

To begin the description, e, is described mathematically by;



Analyzing this figure critically, the figure  severally repeated from the tenth position. This

property is interesting since it goes beyond the expectation of common irrational mathematics

Surname 2

constants; the irrational mathematics constants do not exhibit repetitions at their or patterns after

the decimal place.

The exponential constant is sometimes referred to as the Euler’s number or the Napier

constant. The number was derived by mathematician Jacob Bernoulli when he was studying the

behavior of the compound interest.

The number is very important in the mathematics. It is mostly used in the calculation of the

exponential functions. The number is irrational meaning that is not a ratio of two numbers. The

number is also transcendental meaning that it is not a root of the non-zero polynomial. The value

of e has too many decimal but it is mostly truncated at the 50 decimal place.

The exponential function is also expressed by the use of the Taylor series. The expression is

written as  





















If the value of x is a complex number the value is used in the calculation and the derivation of

the De Moivres theorem. The value of the known digits of the constant e has kept increasing over

time due to the computer technology and the development of the algorithms.

This paper intends to concentrate on fascinating properties of e particularly its natural

growth property. This is accomplished through a basic evaluation of the investment analysis. A

population model of growth is likewise used to clarify the growth property of the exponential

constant. This paper gives the outline that the exponential growth (which has the base of Epsilon)

happens at whatever point the rate of growth, (for example, an increment or reduction) fluctuates

relatively with the size of the original value.

The point where all the logarithms is equal to zero, the slope of the natural logarithm at

that point is specifically equal to one. The logarithms with base higher than e, normally 10 have

slopes less than one. The values of the exhibited give the constant e a unique property. The

constant e is also unique any number close to one has the natural logarithm value equal to the

difference of the number one from the numeric value.

The paper explains the use of the e value in any scenarios and how the vale is important in

various calculations.

Analysis

The value of e is the summations of the expression













Surname 3

The expressions is tabulated as shown below



























 





         

The value of the from the above expression is close to the exact value of the epsilon e. If he

number if the other integers are include into he expression the value will exactly be equal.

Again the graph of the expression 



is drawn; the value of the gradient of the line at the

point 0 is equal to the 2.718. The line of the graph is as in the following graph.

Fig1: Graph of Y against X

The graph gives the exponential curve. The gradient of the graph is given by the value





 . At the point y=1the value of x=0. The inverse of the function of the exponential curve

gives the natural logarithm value e. The inverse of the graph is the graph of the expression y=1/x

1000000

2000000

3000000

4000000

5000000

6000000

7000000

8000000

9000000

0 5 10 15 20 25

Series1

Graph of y against x

Surname 4

The area under the curve which makes the area of unit value is given by the integration of the

expression





from the points 1 to the value e. The area under the curve of the graph indicates the

amount of the space occupied by the space between the x axis, the curve and the limits of the

integration. The value which gives the value of the graph t be equal to the unity value is the one

and the epsilon value. The exponential curve is applicable to many scenarios. The graph is found

in the engineering, economics and the demography analysis.

Again the value  









gives the value of the epsilon if the limit of the expression is infinity.

This is explained is below.

If for example a graph of an investment of a given amount of capital is plotted against the

compounded total amount after sometime in x years the result is exponential growth curve. The

graph plotted is as shown below.

Fig1: Graph diverging to e as y approaches infinity

0 0.5 1 1.5 2 2.5 3

epsilon graph

Surname 5

If the graph is evaluated an assesses, the line of the graph is found that it converges at the

Epsilon constant , e. If the value of x is increased to the infinity, this makes the value of the

outcome to be exactly equal to the 2.718. The expression can be differentiated to give the general

formula of the computing the compound interest.





 









 













  







  



As the value x variable increases, the result of differentiation decreases to null. In

scientific way, it shows that the number of periods, x, becomes unbounded so does expression

 









converge. This shows that the value of the e ;





 









In a case of the business investment, the amount of the loan given by the compound

interest gives approximately the same amount at whichever investment period; weekly, monthly

or quarterly.

This results to the mind to try to prove how the expression explained,  









converge

as x becomes unbounded graphically.

Let the variable y be given by the expression; 



 









Hence the evaluation of the natural log we get;





 











 











We use the L’Hopitals rule to evaluate and differentiate the equation.













 

















Thuse this shows that  thus proved.

Application of the analysis

Surname 6

The concept of the application of the epsilon (e) is found in many areas of study. The natural

logarithm value is found in the in statistics, calculus and the geometry. The following example

gives the application of the exponential function. The common examples apply the use of the

exponential curve in the calculations.

The population estimation

Population growth of living organism obeys the calculus growth. The population growth

may include cells in laboratories, humans in society, wild animals and HIV virus multiplication

in the body. Letting t to denote the time taken for the population growth the population is

represented as a differential equation





. The value of K is called the population growth

rate. The population growth rate is exponential. The exponential growth or decay is however

unrealistic. The rate of growth population is proportion to the initial value of the population

larger population gives a larger population of offspring. The population growth however depends

on factors such as ecology, economics and culture. It is obvious that in a finite planet the

population growth cannot grow continuously indefinitely. The value of the x is a natural

logarithmic function. The function of x is the form of 











where k is a constant and is

the time in years.

The exponential characteristic of the population increment is illustrated during the

original session of population increment in the majority of the natural phenomena. The

exponential growth in most of the living things happens if the rate of change due to population

increment is directly proportional to the instantaneous number of living organisms. Hence,

considering the number of given living organisms to be p, then we have the expression;





Where, 









 = time

 = The rate of increase

The expression is derived from the exponential growth equation. Differentiating the exponential

growth equation; the equation becomes,











Hence;

Surname 7









This makes the variable x be the rate of increase of the. Hence,













Assuming the size of population to be t, then at the time,  



, the size of the

population will have doubled. Thus,





  















With the substation of the above equation to the exponential equation we get,











 





















Thus;











Thus, for the variable x being measured in the percentage form, then the quantity of the

time taken to for the population to double nearly 70 per percentage rate.











Hence for example;

The growth rate of living organisms is 41.99 % per week.

Engineering problems

The Newton’s law of cooling is applied in heat transfer problem is a form of natural logarithm

expression. The temperature model is represented by differential equation. The Newton’s law of

heating states that the rate of heat flow is directly proportional to the area of flow and the

temperature difference between the solid and the fluid boundary. It is an approximate of the

amount of the hat being lost by the cooling body. The variable T denotes the temperature of the

object and T0 the temperature of the ambient environment. In some circumstances the variable t

denotes the measure of time. The equation is thus represented as 



 

K is taken as positive constant. If the Body is at higher temperature than the surrounding, the

value T-T0 is large value; hence  is large and negative. Under these conditions the body

Surname 8

cools rapidly. If the temperature difference is small then the body cools slowly. If a cup of tea is

put in a refrigerator then it cools rapidly. The cooling rate obeys the decay equation which is a

first order differential equation.

The general equation of the Newton’s law of cooling is represented mathematically as

.The constant 



  h is called as the coefficient of heat transfer.

Conclusion

This assessment paper has depicted when the value rate of growth of the population is in

proportion to the magnitude subject. This gives an essential perspective to learn because of thr

applicability of the concept. The concept lays the foundation for the calculation of the rate of

growth of the living organisms and the how to manage some natural phenomena such as the

water hyacinth in water bodies. The new idea can likewise be connected to decide the rate of

increase of immeasurable living organisms, for example, virus in the body tissues.

The value for the epsilon e has been verified to be 2.718281828. The value is useful in

most of the natural logarithm functions as explained above.

Works Cited

Bronshtein, Ilía Nikolaevich, and Konstantin Adol'fovich Semendyayev. Handbook of

mathematics. Springer Science & Business Media, 2013.

Skudrzyk, Eugen. The foundations of acoustics: basic mathematics and basic acoustics. Springer

Science & Business Media, 2012.

Surname 9

Struik, Dirk Jan. A source book in mathematics, 1200-1800. Princeton University Press, 2014.

Place new order. It's free, fast and safe

Paper type

Deadline

Page count

550 words

Total price: $ 15.00

Our customers say

Jeff Curtis

USA, Student

"I'm fully satisfied with the essay I've just received. When I read it, I felt like it was exactly what I wanted to say, but couldn’t find the necessary words. Thank you!"

Ian McGregor

UK, Student

"I don’t know what I would do without your assistance! With your help, I met my deadline just in time and the work was very professional. I will be back in several days with another assignment!"

Shannon Williams

Canada, Student

"It was the perfect experience! I enjoyed working with my writer, he delivered my work on time and followed all the guidelines about the referencing and contents."

Why e is the base of the natural logarithm1-1

Place new order. It's free, fast and safe

Our customers say

Jeff Curtis

Ian McGregor

Shannon Williams