# Logarithmic identities

### Log One Rule

Logarithm of one to any number is zero.

$\large \log_{b}{1} = 0$

### Log of base Rule

Logarithm of base quantity is always one.

$\large \log_{b}{b} \,=\, 1$

### Fundamental Law

$\large b^{\displaystyle \log_{b} m} = m$

### Product Law

Logarithm of the product of two or more quantities is equal to sum of their logs.

$\large (1) \,\,\,\,\,\,$ $\large \log_{b}{(m.n)}$ $\large \,=\,$ $\large \log_{b} m + \log_{b} n$

$\large (2) \,\,\,\,\,\,$ $\large \log_{b}{(m.n.o \cdots)}$ $\large \,=\,$ $\large \log_{b}{m}$ $+$ $\large \log_{b}{n}$ $+$ $\large \log_{b}{o} + \cdots$

### Quotient Rule

Logarithm of quotient of two numbers is equal to difference of their logs.

$\large \log_{b}{\Big(\dfrac{m}{n}\Big)}$ $\large \,=\,$ $\large \log_{b}{m} \,–\, \log_{b}{n}$

### Power Rules

The rules of logarithms which simplify the way to find the values of logarithmic terms by expressing quantity and base quantity of log terms in exponential form.

$\large (1). \,\,\,\,\,\,$ $\large \log_{b}{m^n} \,=\, n\log_{b}{m}$

$\large (2). \,\,\,\,\,\,$ $\large \log_{b^y}{m} \,=\, {\Big(\dfrac{1}{y}\Big)}\log_{b}{m}$

$\large (3). \,\,\,\,\,\,$ $\large \log_{b^y}{m^x} \,=\, {\Big(\dfrac{x}{y}\Big)}\log_{b}{m}$

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### Change of Base formulas

$\large (1). \,\,\,\,\,\,$ $\large \log_{b}{m} = \log_{d}{m} \times \log_{b}{d}$

$\large (2). \,\,\,\,\,\,$ $\large \log_{b}{m} = \dfrac{\log_{d}{m}}{\log_{d}{b}}$

$\large (3). \,\,\,\,\,\,$ $\large \log_{b}{m} = \dfrac{1}{\log_{m}{b}}$

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