# Logarithmic identities

1.

### Log One Rule

Logarithm of one to any number is zero.

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

2.

### Log of base Rule

Logarithm of any number to same number is one.

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

3.

### Fundamental Law

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

4.

### Product Law

Logarithm of the product of two or more numbers to a base is sum of the logarithm of each number to same base.

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

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

5.

### Quotient Rule

Logarithm of quotient of two numbers to a base is the subtraction of logarithm of a number from logarithm of another number.

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

6.

### Power Rule

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

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

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

7.

### Change of Base Rules

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

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

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

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