Quick Review: Logarithm

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Formulas

Any numeral is known as a number. Numbers are of various types. Let us discuss the types of numbers.

  • Definition: ax = b can be represented in logarithmic form as loga b = x
  • log a = x means that 10x = a .
  • 10log a = a (The basic logarithmic identity).
  • log (ab) = log a + log b, a > 0, b > 0
  • log(a/b) = log a -log b, a > 0, b > 0.
  • log an = n (log a) (Logarithm of a power).
  • logx y = logmy / logmx (Change of base rule).
  • logx y = 1 / logy x .
  • logx1 = 0 (x ≠ 0, 1).
  • The natural numbers 1, 2, 3,.... are respectively the logarithms of 10, 100, 1000, .... to the base 10.
  • The logarithm of "0" and negative numbers are not defined.
  • logb1= 0 (∵ b0 = 1)
  • logb b = 1 (∵ b1 = b)
  • y = ln x → ey = x
  • x = ey = → ln x = y
  • x = ln ex = eln x
  • elogb⁡x = x
  • logb by = y
Laws of Logarithm
  • logb⁡MN = logb⁡M + logb⁡N (where b, M and N are positive real numbers and b ≠ 1)
  • logb⁡(M/N) = logb⁡M - logb⁡N (where b, M and N are positive real numbers and b ≠ 1)
  • logb⁡(M) = c logb⁡M (where b, M and N are positive real numbers and b ≠ 1 and c is any real number)
  • logb⁡M = log⁡M/log⁡b = ln⁡M/ln⁡b = logk⁡M/logk⁡b (where b, M and k are positive real numbers and b ≠ 1, k≠1)
  • logb⁡a = 1/loga⁡b (where b, and a are positive real numbers and b ≠ 1, a≠1)
  • If logb⁡M = logb⁡N, then M = N (where b, M and N are positive real numbers and b ≠ 1)
Euler's Number

A mathematical consonant e is the base of the natural logarithm, known as Euler's number. It is also known as Napier's consonant.

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