Logarithm

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Logarithm

  1. 1. LOGARITHM
  2. 2. LOGARITHM OF A NUMBER The logarithm of the number N to the base ‘a’ is the exponent indicating the power to which the base ‘a’ must be raised to obtain the number N. This number is designated as loga N. Hence : loga N x a x N,a 0,a 1&N 0 If a = 10, then we write log b rather than log10 b. If a = e, we write ln b rather than loge b. Here ‘e’ is called as Napiers base & has numerical value equal to 2.7182.
  3. 3.  Remember log10 2 = 0.3010 log103 = 0.4771 ln 2 = 0.693 ln 10 = 2.303 Domain of Definition : The existence and uniqueness of the number loga N can be determined with the help of set of conditions, a > 0 & a 1 & N > 0. The base of the logarithm ‘a’ must not equal unity otherwise numbers not equal to unity will not have a logarithm and any number will be the logarithm of unity.
  4. 4.  Fundamental Logarithmic Identity : a loga N N,a 0,a 1&N 0
  5. 5. THE PRINCIPAL PROPERTIES OF LOGARITHM Let M & N are arbitrary positive numbers, a 0, a 1, b 0, b 1 and is any real number then;i. loga (M N ) = loga M + loga N ; in general loga x1 x 2 ....x n loga x1 loga x 2 ....... log a x nii. loga (M/N) = loga M loga Niii. loga M loga M 1iv. log a M log a M log a Mv. Changing of base logb M = log a bNote :loga 1 = 0 loga a = 1log1/a a = -1 logb a = 1/loga b ax=exlna
  6. 6. LOGARITHMIC EQUATIONThe equality loga x = loga y is possible if and only if x = y i.e. loga x loga y x yAlways check validity of given equation, x 0, y 0,a 0,a 1
  7. 7. GRAPHS OF LOGARITHMIC FUNCTIONS (y = loga x) y y a>1 0<a<1 (1, 0) x x (1, 0) If the number and the base are on the same side of the unity, then the logarithm is positive. If the number and the base are on the opposite sides of unity, then the logarithm is negative.
  8. 8. Graph of y = loga | x | y y 0<a<1 a>1(-1, 0) 0 (1, 0) x (-1, 0) (1, 0)
  9. 9. PROPERTIES OF LOGARITHM INEQUALITIES Constant Base x y 0 if a 1i. loga x loga y 0 x y if 0 a 1 x ap if a 1ii. loga x p x ap if 0 a 1
  10. 10.  Variable Base For variable base a logarithmic inequality is solved by reducing it to a set of seven inequalities as follows : loga x log x y a 1 & x y 0 OR 0 a 1&0 x y Note : In order to apply the above concepts of solving logarithmic equations and inequations, first make the bases of logarithms equal to both sides.

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