# Logarithm

## Definition of Logarithm

The logarithm is the inverse function of exponentiation. The logarithm of a positive real number x with respect to base b is the exponent by which b must be raised to produce x.

## Symbol of Logarithm

The Logarithm of x to base b may be expressed as logb(x) and pronounced as the logarithm of x to base b.

$y = {\log _a}x\,\,if\,and\,only\,if\,x = {a^y},a > 0,a \ne 1$

## Some Logarithmic Identities or Logarithmic formulas

1. ${\log _a}1 = 0$
2. ${\log _a}a = 1$
3. ${\log _a}xy = {\log _a}x + {\log _a}y$
4. ${\log _a}\frac{x}{y} = {\log _a}x - {\log _a}y$
5. ${\log _a}\left( {{x^n}} \right) = n{\log _a}x$
6. ${\log _a}\sqrt[n]{x} = \frac{1}{n}{\log _a}x$
7. ${\log _a}x = \frac{{{{\log }_c}x}}{{{{\log }_c}a}} = {\log _c}x.{\log _a}c\,\,\,\,\,\,\,\,\,where\,c > 0,\,\,c \ne 1$
8. ${\log _a}c = \frac{1}{{{{\log }_c}a}}$
9. $x = {a^{{{\log }_a}x}}$

## Logarithm to Base 10

1. ${\log _{10}}x = \log x$

## Natural Logarithm

1. ${\log _e}x = \ln x\,\,\,\,\,where\,\,\,e = 2.717281828….$
2. $\log x = \frac{1}{{\ln 10}}\ln x = 0.434294\,\ln x$
3. $\ln x = \frac{1}{{\log e}}\log x = 2.302585\,\log x$