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
- elogbx = x
- logb by = y
Laws of Logarithm
- logbMN = logbM + logbN (where b, M and N are positive real numbers and b ≠ 1)
- logb(M/N) = logbM - logbN (where b, M and N are positive real numbers and b ≠ 1)
- logb(Mc ) = c logbM (where b, M and N are positive real numbers and b ≠ 1 and c is any real number)
- logbM = logM/logb = lnM/lnb = logkM/logkb (where b, M and k are positive real numbers and b ≠ 1, k≠1)
- logba = 1/logab (where b, and a are positive real numbers and b ≠ 1, a≠1)
- If logbM = logbN, then M = N (where b, M and N are positive real numbers and b ≠ 1)
A mathematical consonant e is the base of the natural logarithm, known as Euler's number. It is also known as Napier's consonant.