# Laws of Logarithms

**Introduction**

Three most important Laws of Logarithms Rules of logarithms are:

- Rule of Multiplication
- Rule of Division
- 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 $${\mathrm{log}}_{a}XY{\text{=log}}_{\text{a}}X{\text{+log}}_{\text{a}}Y$$ |
$${\mathrm{log}}_{a}\left(2\times 3\right){\text{=log}}_{\text{a}}2{\text{+log}}_{\text{a}}3$$ |

2 | Rule of Division $${\mathrm{log}}_{a}\frac{X}{Y}{\text{=log}}_{\text{a}}X{\text{-log}}_{\text{a}}Y$$ |
$${\mathrm{log}}_{a}\left(\frac{4}{5}\right){\text{=log}}_{\text{a}}{\text{4-log}}_{\text{a}}5$$ |

3 | Rule of Power $${\mathrm{log}}_{a}{X}^{n}{\text{=nlog}}_{\text{a}}X$$ |
$${\mathrm{log}}_{a}{\text{7}}^{2}=2{\mathrm{log}}_{a}7$$ |

Table 1

Three other laws of logarithms are stated as below.No. |
Rules of indices |
Examples |

1 | Logarithm equal to 1 $${\mathrm{log}}_{a}a\text{=1}$$ |
$${\mathrm{log}}_{5}\text{5}=1$$ |

2 | Logarithm equal to 0 $${\mathrm{log}}_{a}1\text{=0}$$ |
$${\mathrm{log}}_{2}1=0$$ |

3 | $${\mathrm{log}}_{a}\frac{1}{X}{\text{=log}}_{\text{a}}{X}^{-1}{\text{=-log}}_{\text{a}}X$$ | $${\mathrm{log}}_{2}\frac{1}{7}={\mathrm{log}}_{2}{7}^{-1}=-{\mathrm{log}}_{2}7$$ |

Table 2

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 |

$${\mathrm{log}}_{a}b$$ | $${\mathrm{log}}_{a}b=\frac{{\mathrm{log}}_{c}b}{{\mathrm{log}}_{c}a}$$ |

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