Logarithms are used to simplify the complex mathematical equations and help the students to solve such equations and simply their mathematical expressions. Logarithms are closely associated to the exponential terminologies and therefore the properties of logarithms are similar or linked with the exponent properties. The properties of logarithms assist the individual to simplify or shorten the complex and long mathematical expression. Logarithms are the examples for the concave functions that was introduced or identified by the scientist, John Napier, during the early 17^th century that was aimed to simply the complex mathematical expression unit simpler calculations.

There are various rules that is used while solving any mathematical expression using logarithms and hence are known as laws of logarithms. The law of logarithms are used and practiced for writing the logarithms in different types of problems. The law is applicable and applied to logarithms irrespective of any form of base but in case of the similar base, the logarithms must be used and practiced throughout the calculation and the detail steps should also be exhibited that will explain the application of logarithms. This laws of logarithms is useful for providing every individual the detail procedure to add or calculate the two logarithms collectively.

Logarithms are used for inversing the effect of exponents other than being an inverse operation. The slide rule is applicable and practiced by either adding or deducting the logarithms that is equal to the method of division and multiplication. Natural algorithms are considered as the most beneficial method for solving problems as it is used or practiced to solve the mathematical problems related to half-life, unknown time and decay constant within the exponential decay mathematical problems. Logarithms are beneficial in various branches or fields related to science and mathematics that is furthered used for solving finance-related problems that comprise or include compound interest.

There are different logarithmic properties that is used for solving the mathematical expression and are classified as quotient rule, product rule, Change of Base rule and power rule. The mathematical expression of each of the logarithmic property are as follows:

• Product rule: logb(xy) = logbx + logby
• Quotient rule: logb(x/y) = logbx – logby
• Power rule: logb(xr) = rlogbx
• Change of base rule: logb1 = 0

The logarithm of the quotient or the product is defined as the total or difference of logarithms and the logarithm of a number to its power is defined as the power intervals the logarithm of that particular number.

The below example will help the reader to understand the use of logarithm properties in an appropriate way. It is noted that each mathematical problem has its individual method of solving and it is not compulsory that every problem will have the similar approach of solution. Therefore, the below mentioned examples will help the readers to understand the aspect of solving the exponential mathematical question.

1. log₃ (9x) = log₃ 9 + log₃ x (we will use the product rule for solving this mathematical problem)

log₃ 9 + log₃ x = log₃ 3² + log₃ x = 2 + log₃ x (In this particular step, the primary focus will be to simplify each individual addend and also notice that log₃ 9 can be simplified but the component log₃ x cannot be simplified. The next step will be re-write log₃ 9 in the form of log₃ 3², and then utilize the property of the logarithm log_b b^x = x. The other step that can be used or utilised is to simplify log₃ 9, by converting log₃ 9 = y to 3^y = 9 and discovering that y = 2)

Therefore, log₃ (9x) = 2 + log₃ x.

1. log₃ 9⁴ = 4 log₃ 9 ( we will use the power rule for solving this mathematical problem, the value of 9⁴ can be calculated but doing so will not make any difference or make the problem easier or simplified. Hence, the power rule of logarithm is used and the value of log₃ 9⁴ is re-written in the form of 4 log₃ 9)

4 log₃ 9 = 4•2 (In this step, we can recognize that as 3² = 9, therefore, the value of log₃ 9 = 2)

Therefore, 4 log₃ 9 = 8.

It is clear that logarithm is considered as the opposite or alternative of power. For example if we consider the base with b=2 and increase the power denoted as k= 3, we will consider the mathematical expression as 2³. The ultimate result that will be extracted from this expression will be denoted in the form of c that will be further defined as 23=c. The exponentiation rule will be applied to calculate the result extracted from this mathematical expression as:

c=2³=8

If the value of exponent denoted as “k” is not known and only the base value of b=2 is provided them the total value or result of the exponent c will be 8. In order to calculate the value of the exponent k, type mathematical expression used will be in the form of 2k=8. Hence, it is clearly stated from the above explanation that logarithm is defend as the function that is solely responsible for carrying out any mathematical expression as it is generally denoted as “log” or  “log base 2”. The log base 2 is denoted and defined using the below expression:

Log₂c=k, and is considered as the solution for the mathematical problem denoted in the form of 2k=c, if the value of the number c is already provided. Hence, the basic rules and regulations of using the logarithms are clearly explained and how can this function be derived from the exponential or inverse function is highlighted. Each logarithm rule has an individual formula that is used and applied in solving any particular mathematical expression. It is also stated that each rule of logarithms can be derived and used from the standard expression as it is not mandatory that each formula is present sometimes it is compulsory to derive the formula and base the mathematical problems on those derived formulas.