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.
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.
Fundamental Logarithmic Identity : a loga N N,a 0,a 1&N 0
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
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
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.
Graph of y = loga | x | y y 0<a<1 a>1(-1, 0) 0 (1, 0) x (-1, 0) (1, 0)
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
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.