Logarithms are an alternative method of representing exponents in the mathematical field of mathematics. Another number may be expressed as an equivalent value using the logarithm of a number with a base. The logarithm is the mathematical function that is the opposite of the exponentiation function.
For example, if 102 = 100,
Then log10 100 = 2.
As a result, you can assume that,
LogbX = n or bn = X Where
b = base of the logarithmic function
History Of Logarithms
In the 17th century, John Napier was the first person to present the idea of logarithms. In later years, a wide variety of professionals, including scientists, navigators, engineers, and others, utilized it to simplify numerous computations. Exponentiation may be considered the forward process, while logarithms can be regarded as their inverse.
Previously, Napier would manage his relationships using base 1/e. Because of this, some individuals refer to logs (or ln x) as the Napierian Logarithm. In the beginning, Napier referred to his structure as “artificial numbers;” it wasn’t until much later that they began to be dubbed logarithms.
Henry Briggs (1561–1630), a mathematician and Oxford professor, introduced Europe to logarithms. Briggs also began working with base 10, which was gaining popularity then. He also released several logarithm tables, ranging from one to twenty thousand and ninety to one hundred thousand. Additionally, Briggs is the one who standardized the use of the word “mantissa.”
Edmund Gunter, who lived from 1581 to 1626, is credited for publishing the logarithm tables for sines and tangents. To pique your curiosity, Gunter is the one who combined the phrases cosine and cotangent.
Understanding the Concept of Logarithms
A logarithm may be considered the power to which a number has to be raised to get a particular set of values. Large numbers may be expressed in the most straightforward manner using this method.
A logarithm has a variety of significant qualities that demonstrate the multiplication and division of logarithms and may also be signified in the form of logarithms of addition and subtraction. These properties are shown by the fact that logarithms can be added and subtracted.
The logarithm of positive real numbers a concerning base b, which is positive real numbers that are not equal to 1, is the exponent that must be increased to get a from b.
Usage of Logarithm
Logarithms are a way that enables more complicated mathematical operations such as multiplication, division, and the calculation of a root to be done utilizing addition and subtraction. This is made possible because logarithms provide a mechanism for doing so.
Every single number may be written down in what we today refer to as an exponential form, which means that eight can be written as 2, 25 can be written as 5, and so on.
The fact that logarithms reduce the more complicated operations of multiplication and division to the more simple actions of addition and subtraction makes them so valuable. When huge numbers are represented as logarithms, the process of expansion is transformed into the acquisition of exponents.
Calculations are made more easily and quickly by using logarithms. Using them, you may significantly reduce the amount of time required for multiple significant amounts.
Scientists can compute the product of two separate integers, x, and y, by first determining the individual logarithms of each number, adding those logarithms together, and then consulting a table to get the precise number corresponding to the logarithm that just calculated.
This property is referred to as its antilogarithm. Logarithms are governed by a set of laws and regulations which provide direction and assistance to individuals who are using logarithms. Some many distinct rules and regulations are related to logarithms, and each table will have its own set of them.
Types of Logarithms
The majority of the time, we will always be dealing with two distinct kinds of logarithms, which are:
- Common Logarithm
- Natural Logarithm
Common Logarithm
The common logarithm is also referred to as the logarithms of base 10. It is denoted by the symbol log ten or logs for short. As an illustration, the common logarithm of 1000 is represented on paper as a log (1000). The common logarithm answers the question of how many times we need to multiply the number 10 to achieve the desired outcome.
As an example, log (100) = 2
The result of multiplying the number 10 by itself twice is the number 100.
Natural Logarithm
The logarithm using e as its basis is known as the natural logarithm. Both ln and loge are acceptable representations of the natural logarithm. In this context, “e” refers to the Euler constant, which has a value roughly equivalent to 2.71828. For instance, the natural logarithm of the number 78 is represented by the symbol ln 78. The natural logarithm tells us how many times we must multiply “e” to acquire the necessary output.
For example, ln (78) = 4.357.
Therefore, the answer to the base e logarithm problem for 78 is 4.357.
Application of Logarithms
Logarithms have a wide variety of real-world applications and illustrative examples. Finding a solution to issues involving exponentiation is one of the principal uses of this method. Henry Briggs compiled the table of logarithms in 1617, which is a resource that may assist persons in completing the stages involved in solving logarithmic mathematical issues.
This occurred not long after Napier’s discovery, although this table employed ten as its foundation instead of 1. Briggs’s initial table includes the fundamental logarithms of all numbers ranging from one to one thousand.
Logarithms were used to create the slide rule, a pair of split scales utilized for calculation. Through the use of number lines, issues may be solved using a slide rule, even in the absence of a calculator.
Logarithms were included in some different laws and philosophies. A few good examples are the theory of probabilities and the rules that govern how coin flips should be conducted. The law of the iterated logarithm determines the ratio of heads to tails while doing a coin flip.
A Real-Life Example of Logarithms
Several particular instances of logarithms are used in reality in the real world.
Earthquakes
The Richter scale, used to control earthquake severity, includes logarithmic functions as an integral component. This is mainly associated with the quantity of energy released during the quake. Seismographs are instruments that measure movement on the earth. As a result, of the detail that the scale uses a logarithmic method, a tenfold rise in amplitude corresponds to each whole-number increase in the earthquake’s magnitude.
Sound Decibels
Logarithms may also be used in decibel measurements, which is another example. According to the website Physclips, a decibel is a logarithmic unit used to measure sound’s intensity. Additionally, it has widespread use in electronics and communication research.
pH Balances
In chemistry, logarithms may be used in the process of deciding the acidity or alkalinity of a variety of substances. Worksheets are available that illustrate the various chemicals as well as the logarithmic formulae that are used in the calculation of pH.