Exploring Logarithms: The Math Behind Multiplication and Division - Essay Sample

Introduction

A logarithm is an exponential in which the base must be raised to generate a particular number. Logarithms are utilized in various calculations in science and engineering, economics, and business. Before the invention of the calculator, logarithms were used in the multiplication process by substituting the multiplication operation with addition. Again, the division operation was replaced by subtraction. Logarithms are still vital in various ways through the underlying assumption of the logarithm function (Borwein et al., 2017). Currently, most of the learners in different levels of the learning institutions experience a hard time trying to understand the logarithm concepts. The work discusses the history of logarithms, chooses, and justifies a resource utilized in demonstration of laws of logarithms, explains and designs through interactive simulation, and reflects on how mathematical laws manage the problem-solving process.

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Investigation

History of logarithms

The history of logarithms is a narration of a correspondence between the addition of the real number line and the positive real numbers which were formalized back in the seventeenth century. It had a wide range of applications, including simplifying calculations until the initiation of digital computers. John Napier and Joost Burgi, a Scotsman and a Swiss respectively, is the independent inventor of logarithms. Publishing of logarithms by Burgi was done in 1620, while that Napier was published in the year 1614 (Quintanilla, 2018). The logarithms that they invented varied from each other and from the natural and common logarithms that are in use today. Burgis utilized the geometric approach, while Napiers style used an algebraic approach.

The main aims of the two inventors were to make mathematics calculations simpler. Neither of these men came up with the concept of a logarithmic base. According to Napier, logarithms are defined as the ration of two lengths in a geometric form. Therefore, it is different from the present definition version that identifies it as exponentials. John Bernoulli and john Walis recognized the chances of defining logarithms as exponents in the year 1694 and 1685. The combined efforts of Henry Briggs and Napier in the year 1624 resulted in the invention of the ordinary logarithm system. (Weber, 2019) The natural logarithms’ began as more or less accidental differences of original logarithms of Napier. Their real importance was not realized until later. The first natural logarithm happened in the year 1618. Publishing of the logarithm tables was done in several forms over the past four centuries.

Again, the logarithm idea was utilized in the construction of slide rule that became ever-present in engineering and science until the 1970s. A logarithm is of considerable significance because it is applicable in various settings like finance; astronomy (Borwein et al., 2017). The search for an area expression against the rectangular hyperbola was the advancement in the creation of the natural logarithm. It necessitated the assimilation of new functions into ordinary mathematics. Euler is another pioneer of the logarithms’ since he defined natural logarithms and exponential functions. He wrote a textbook in the year 1748 in which he published the current standard approach to logarithms through the inverse function.

Features of Logarithms

The scientists accepted logarithms because they had several functional features that simplified tedious calculations, for instance, scientists could solve a problem of finding the product of two digits by looking on the logarithm of each of the numbers in the table. They could sum up the logarithms and then using the table again to find the antilogarithm resulting in the final number. Practically, this means that common logarithms can be expressed in the relationship log mn=log m + log n. for example, calculation of 100 x 1000 can be found by searching at the logarithms’ of 1000 which is 3 and that of 100 which is 2.Adding the logarithms results to 5. After this, the antilogarithm is determined to find the final answer, which is 100,000 from the table. Similarly, the problems on division can also be converted into subtraction with logarithms; log m/n = log m- log n. Additionally, the roots and powers can be simplified using logarithms (Borwein et al., 2017).

Some of the law of logarithms entails;

log m/n = log m- log n

log mn=log m + log n

log b n = log a n log b a

log b n p = p log b n

log b a√n = 1/a log b n

Choice and Justification

The resource that I have selected is a video titled laws of logarithms. It is a short video that demonstrates the various laws of logarithms that are crucial in solving problems. It provides examples of the applicability of the laws in calculations, thus making it more practical. I believe this is a resource that uses simple language and terms to explain the concepts. Additionally, the laws can easily be understood with ease because examples are also provided, resulting in an in-depth knowledge concerning the laws of logarithms. The link to this short move is

https://www.youtube.com/watch?v=ulAxVjqAhRsExplanation and Design

Below is a link to my designed interactive simulation.

https://bit.ly/2AtxZwG

Reflection

Problem-solving offers a working foundation in the application of mathematics. The mathematics problem gives opportunities to extend and solidify what they understand. The problem solving has four steps that entail understanding the problem, devising, and carrying out plans. Looking back is another vital step to ensuring that every concept is well utilized to find the solution (Borwein et al., 2017). A real problem-solving problem enables learners to be intuitive, flexible, and creative. The laws of logarithms allow rewriting of several expressions that entail logarithms in diverse ways. The laws are applicable to logarithms of any base. However, similar bases must be utilized in all calculations. The law informs us of the addition of two logarithms together (Weber, 2019).

Conclusion

In summary, the laws of logarithms are of considerable significance because of the various ranges of applications in many fields that entail science and engineering, economics, business, among others. Before the invention of calculators, logarithms were utilized in solving multiplication and division processes. Logarithms history is a narration of a connection between the addition of the real number line and the positive real numbers. These were formalized back in the seventeenth century. John Napier and Joost Burgi are the independent discoverers of logarithms. Burgis used the geometric approach while, on the other hand, the Napiers style was algebraic. The formation of the natural logarithms was because of the exploration of expression against rectangular hyperbola, thus necessitating the assimilation of new functions into ordinary mathematics. The scientists accepted logarithms because they had several functional features that made tedious calculations simpler.

References

Borwein, J. M., Hare, K. G., & Lynch, J. G. (2017). Generalized continued logarithms and related continued fractions. Journal of Integer Sequences, 20(2), 3.

Quintanilla, J. (2018). My Favorite Lesson: Developing Intuition for Logarithms. Mathematics Teacher, 112(1), 80-80. https://www.jstor.org/stable/10.5951/mathteacher.112.1.0080Retrieved from https://www.youtube.com/watch?v=ulAxVjqAhRs

Weber, C. (2019). Making Sense of Logarithms as Counting Divisions. The Mathematics Teacher, 112(5), 374-380. https://www.jstor.org/stable/10.5951/mathteacher.112.5.0374?seq=1