Unlocking the Power of Natural Logarithms

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This paper focuses on the natural logarithm of numbers in mathematics. In this type of logarithm, the base is a mathematical e, in which e is a transcendental and an irrational number that can be evaluated as 2.718281828459 and approximated as 2.718. Research shows that the concept of natural logarithm was developed by Alphonse Antonio de Sarasa and Gregoire de Saint Vincent as early as 1649 using the quadrature of the hyperbola through the equation xy = 1 (Euler 27). These mathematicians attempted to provide a solution to the area of the sectors of the hyperbola, which emerged as a hyperbolic logarithm function.  Nicholas Mercator further developed the work of Alphonse Antonio de Sarasa and Gregoire de Saint Vincent in his scholarly work Logarithmotechnia that was published in 1668. Euler (p.34) further states that it was in Mercator’s work that this concept was first termed as ‘natural logarithm’. The natural logarithm is given by the notation ln x or loge x while being used in various programming languages and scientific contexts (Kahn and David 53). This function derived its name from a mathematical evaluation of the area and arcs of circular and hyperbolic functions. The connection between these functions demonstrated the “naturalness” of this logarithm; hence the name. Natural function can also be regarded as the inverse of the exponential function in mathematical evaluation.


Figure 1: Hyperbolic sector in which one area indicates Euler's number

         Natural logarithm has similar properties to the conventional logarithms. The significant difference between natural logarithm is its base that is evaluated to a base of 2.718 that is given by letter e. The exponential growth exhibits other log-like properties, such as the addition, multiplication, and division laws. Calculus is the study of change, specifically by studying instantaneous changes and applying them to a domain (Kahn and David 57). One relation between changes that can be studied and used to analyze, evaluate and predict results is the exponential growth or decay relation, which occurs when a variable, mathematically changes in a relationship that is proportional to itself. This type of growth can be studied to draw predictions based on another variable.

            Exponential growth can be utilized for many calculations primarily involving interest compounding, population growth, economic growth, etc. Any function that has an exponential relation can be studied by using this concept. Deep understanding of the origin of the exponential constant is the most appropriate starting point.

         The letter ‘e’ in mathematics stands for the exponential constant. It has an approximate value of 2.718. Its appearance frequency in physical and economic mathematics has made it become a known constant. A variable that was historically first encountered in a compound interest problem, but it was Jacob Bernoulli who analyzed this problem and by using the binomial theorem was able to approximate e in between 2 and 3. This was the first time a number was defined by a limiting process, but it was not precise enough. Several years later Leonhard Euler published Introductio in Analysin Infinitorum, in which he gives a full notion of the concept of e and approximates it to eighteenth decimal places (Maor). From this exponential constant, we will derive the exponential growth formula and elaborate on different uses.

            Continuing with the assumption that the size of the variable is growing or decaying in proportion to itself, we can write: dx/dt = Kx , in which dx/dt represents the growth rate, K symbolizes the relative growth frame and is negative if its decaying, and t is the independent variable, usually representing time. Simplifying we obtain dx/x = (K * x * dt)/x, and after integrating ∫ 1/x dx = ∫ Kdx we obtain lnx = Kt + C where C is the general constant in integration. After the simplification of e on the assumption that C is a general constant lnx = eKt+C we obtain X(t) = Cext . Finally, we will remove the constant C by assuming its initial value X (0) to represented as X0 (Subzero), leaving the final formula X (t) = X0ekt. This is the derivations of the general formula for exponential growth and it can be used to analyze samples and make predictions for future values. A real-life example to illustrate this concept involving economic growth is the following:

Mark has $5,000 in his savings account. If his account gives him 2% every year, how much will he have in 5 years?

Using the formula that we derived, we can write X (t) = (5000) × e0.02×5 which equals approximately 5525.85.

            In this case, the formula is taking into account the exponential relation that occurs when the percentage of additional money becomes bigger since so does the value of money with time.

This key differentiates this type of growth, which is the consideration of its growing value with respect to another variable, in this case, time. “Growth in growth” is a simple phrase that describes this relationship (Yuri p.23). An analytical approach example with exponential decay is where the relative growth rate has to be determined to then be used to calculate future values, such as:

The recorded native population of a town in Malaga, Spain called Marbella in 2000 was 12,000 habitants. The recorded amount 5 years later was 10,200 habitants. What will be the population in 2022?

First, we will calculate the relative growth rate by plugging in values to X (5) = (12000) × e5k

and substituting X (5) for the correspondent given value of 10,200.

10200 = 12000 × e5k by solving this we conclude that K = −0.0325

After we have calculated the relative growth rate, we can find the future predicted value by plugging in values to the formula X (t) = X0ekt, leaving X (t) = 12000×e-0.0325×22 which yields approximately 5870 individuals by 2022.

            A more complex example of the application of this function in Calculus is the following:

Electricity leaving a battery at a rate that is proportional to the voltage as dv/dt = − (1/10) v

Here the rate of change is given, so we isolate the variables correspondingly to obtain dv/v = − (1/10)dt and after integration and simplification, v = v0 × e−t/10  . These last steps are elaborated on the previous example and the solution reflects the voltage as a function of time that is dependent on t and proportional to its value.

This linear graph represents the voltage with an initial value of 100 Volts in relation to time in minutes over an hour frame. This graph can easily help identify what is known as the exponential pattern, in this case, exponential decay.

            There are endless daily applications where the study of exponential growth not only plays

a big role in analysis but sometimes can even be helpful when making personal decisions. Such include mortgages calculations, population change, resource availability, monetary growth, and even estimate specific prices according to time (Yuri 38). The exponential growth formula is a notable factor in Calculus that can be modified to work with different types of growth relations that might involve more independent variables or more complex intervals. By using the constant ‘e’ we can take the assumption that a variable changes proportionally and study its relative grow/decay rate and possible future values, allowing for very exact calculations on everyday life situations and simultaneously helping understand the concept of ‘e’ (Koni︠a︡gin and Shparlinski 156).

Works Cited

Euler. Introduction to Analysis of the Infinite: Book I. New York, NY: Springer New York, 1988. Internet resource https://books.google.co.ke/books?id=r96RS-3RuE8C&dq=exponential+functions&hl=en&sa=X&ved=0ahUKEwi53dyzpInaAhWGPhQKHUH1BV4Q6AEIMjAC

Kahn, David. Attacking Problems in Logarithms and Exponential Functions. , 2015. Print.             https://books.google.co.ke/books?id=xQpmCgAAQBAJ&printsec=frontcover&dq=exponential+functions&hl=en&sa=X&ved=0ahUKEwi53dyzpInaAhWGPhQKHUH1BV4Q6AEIJzAA#v=onepage&q=exponential%20functions&f=false

Koniagin, S V, and Igor E. Shparlinski. Character Sums with Exponential Functions and Their Applications. Cambridge: Cambridge University Press, 1999. Print.

Maor, Eli. E: The Story of a Number. Princeton, N.J: Princeton University Press, 2009. Internet resource.

Shestopaloff, Yuri K. Properties and Interrelationships of Polynomial, Exponential, Logarithmic, and Power Functions with Applications to Modeling Natural Phenomena. Toronto: AKVY Press, 2010. Print.

September 25, 2023


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