# The Logarithm as the Basis and Foundation of Mathematics

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Logarithms
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Introduction
Logarithms entail expressing numbers using notions like log or in. They are used in wide area of applications since many areas that require mathematical concept such as chemistry and physics require the concept of logarithms. Many people argue that logarithms are complicated but they are easy to understand how they work. Logarithms have many uses in real life situation but many people do not understand their application since they limit themselves to mathematical concept. Generally, the paper will discuss in details on how logarithms work and how they can be applied.
How logarithms work
Logarithms work using a simple concept which should be followed strictly; this simple concept is referred to use law of logarithms. Understanding law of logarithm entails understanding how logarithm works. This section will explain 5 major laws of logarithm and by the end of the section the whole concept of how logarithm works will be clear. However, before tackling the laws, it is good first to explain some basics of logarithms that are very important. To start with, when a number is written in this form log45 or log2 it means that the number is to base ten which is referred to use log to base ten. Other logarithms are usually specified on their bases like log24 which means that it is log4 to base two. Finally, there is another log which is referred as natural log which is written using initial (In). IT is also expressed as log to base written as loge. The following section explains how logarithm works using laws of logarithms.
The first law states that if A & B are positive real numbers, then logAB= logbA +logbB. This law can be perfectly proved in order to make the whole concept clear and below is description.
Let logaB=m and logbB= n
In index form, logbA=m A=bM
AB= bm x bn = bm+n………………….logbAB=m+n
LogbAB=logbA + logbB. This is the most common law that is widely used to deal with exponents in mathematics. When applied in mathematics, the law acts when multiplying two numbers with two similar bases. In this case, their powers are added instead of being multiplied. This is the first law of logarithms that explain how they work.
The second law of logarithm states that logb (A/B)= logbA- log bB. This law applies mainly when it comes to division. It means that if you dividing two numbers with the same base, you should subtract their exponents in order to get the value. This is a simple way of solving numbers with huge powers using logarithm.
The third law that explains how logarithms work is the law that states that logb(A)n = nlogbA. This is also a common law that clearly explains how logarithms work. The law applies when solving mathematical calculations with more than one power. For instance, when one wants to square a number which is already raised to a power, one can apply logarithms and the problem can be solved easily as indicated through the initials. This is the main logarithmic concept that explains how it works.
The fourth law that explains how logarithm works states that loga1=0 if a not equal to 0. This means that if any number is raised to 1 is equal to 0 on the condition that (a) is not equal to 0. This concept is used to simplify mathematics where one is made to understand that any log raised to power 1 is equal to 0 and thus there is no need of calculations.
Finally, the last law states that logaa=1 on a condition that (a) is not equal to zero. This is a wide concept that is used to explain how logarithm works in cases of a similar base and power. In such cases the result is usually 1 since it means a number when raised when raised to one will only go one time. This concept also simplifies mathematics in a greater way.
How to use logarithms
There are many ways in which logarithms are applied. To begin with, logarithms can be used to deal with big numbers. For instance, it is very hard to calculate big numbers but through the use of logarithm this can be made simpler. When simplifying big numbers, logarithm to base 10 is used. The following examples will explain this in details, 102=100, 103=1000, 104= 10, 000. This is logarithmic and indices concept that is used to deal with huge numbers.
In calculations, logarithms are made simpler to use through the use of calculators. When solving such as this, log100, one is supposed to feed the problem directly to the calculator since calculators offers a direct solution to all problems involving logarithm to base 10. Any number that is not under any base is assumed to be in base 10 and it is usually feed directly to the calculator. This concept explains how to use the simplest form of a logarithm in any problem.
Logarithms to base e which are usually referred as natural log are also simple to work with and solve any kind of a problem. This is because calculators usually calculate problems involving natural log directly without any need for formulations. For instance, when solving a problem like this, In23, one is supposed to feed the calculator directly and get the solution. This is also a simple use of logarithms. However, there are some situations that require critical evaluation and manipulation of numbers in order to get the right solution.
When solving logarithmic problems that are not either to base 10 or to base e, the concept of indices is applied. This is where the log sing is removed and the exponents are interchanged. The problem can be solved in the following ways.

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Surname 1

Logarithms
Name:
Professor:
Course:
Institution:
Date: