Table of Contents

## 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 *log _{b}(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

- {\log _a}1 = 0
- {\log _a}a = 1
- {\log _a}xy = {\log _a}x + {\log _a}y
- {\log _a}\frac{x}{y} = {\log _a}x - {\log _a}y
- {\log _a}\left( {{x^n}} \right) = n{\log _a}x
- {\log _a}\sqrt[n]{x} = \frac{1}{n}{\log _a}x
- {\log _a}x = \frac{{{{\log }_c}x}}{{{{\log }_c}a}} = {\log _c}x.{\log _a}c\,\,\,\,\,\,\,\,\,where\,c > 0,\,\,c \ne 1
- {\log _a}c = \frac{1}{{{{\log }_c}a}}
- x = {a^{{{\log }_a}x}}

## Logarithm to Base 10

- {\log _{10}}x = \log x

## Natural Logarithm

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

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