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
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