Laws of Logarithms

Introduction
Three most important Laws of Logarithms Rules of logarithms are:

1. Rule of Multiplication
2. Rule of Division
3. Rule of Power 

Three important rules of indices are listed in table 1. Rules of multiplication and division are applied for logarithms containing identical base.

No.	Rules of Logarithm	Examples
1	Rule of Multiplication $${\log }_{a}XY{\text{= log}}_{\text{a}}X{\text{ + log}}_{\text{a}}Y$$	$${\log }_a\left(2\times 3\right){\text{= log}}_{\text{a}}2{\text{ + log}}_{\text{a}}3$$
2	Rule of Division $${\log }_a\frac{X}{Y}{\text{= log}}_{\text{a}}X{\text{ - log}}_{\text{a}}Y$$	$${\log }_a\left(\frac{4}{5}\right){\text{= log}}_{\text{a}}{\text{4 - log}}_{\text{a}}5$$
3	Rule of Power $${\log }_a{X}^n{\text{ = nlog}}_{\text{a}}X$$	$${\log }_a{\text{7}}^2=2{\log }_{a}7$$

Table 1

Three other laws of logarithms are stated as below.

No.	Rules of indices	Examples
1	Logarithm equal to 1 $${\log }_{a}a\text{ =1}$$	$${\log }_5\text{5}=1$$
2	Logarithm equal to 0 $${\log }_{a}1\text{ =0}$$	$${\log }_{2}1=0$$
3	$${\log }_a\frac{1}{X}{\text{ = log}}_{\text{a}}{X}^{-1}{\text{ = -log}}_{\text{a}}X$$	$${\log }_2\frac{1}{7}={\log }_2{7}^{-1}=-{\log }_{2}7$$

Table 2

Changing the base of logarithm

• Consider the following logarithm with base a and law used to change the base to base c. 
• The new base is c with two new logarithms are in division operation.

From base a	To base c
$${\log }_{a}b$$	$${\log }_{a}b=\frac{{\log }_{c}b}{{\log }_{c}a}$$

Table 2