GETTING STARTED
Let us say ax = b, where b > 0, a> 0 and a ≠ 1.
☛ The exponent x to which base a is raised to obtain a number b is called ‘logarithm of b to the base a’
Eg. 24 = 16
Hence log 2 16 = 4
☛ The base of the logarithm, a, has to be a positive real number. Therefore b will also be a positive number.
☛ Many a times, a logarithm will be written without a base. Eg. log (125) = log 10(125) = 2.096
It means that the base is 10, and it is called as ‘common log’.
☛ Another expression used frequently is ln (x). It denotes natural log where the base is e (Euler’s number) – 2.71828.
Eg. Ln (7.389) = 2
Properties of Logarithms:
(i)log x x =1
Eg. Log 2424 = 1
(ii) log a(x. y) = log a x + log ay
Eg. log 232 = log 2(16 x 2)
= log 216 + log 22 = 4+1 = 5
(iii) log a(x/y) = log ax – log ay
Eg. log 2(32) = log 2(64/2)
= log 264 – log 22 = 6 – 1 =5
(iv) log a1 = 0
Eg. log 101 = 0
(v) log a(xp)= p log a x
Eg. log 2(32) = 5 log 2(2)
= 5 x 1 = 5
(vi) log ax = 1/ log xa
Eg. Let us consider log 322
Log 322 = 1/log 232 = 1/5
(vii) log ax = log bx/log ba = log x/log a
Eg. Log 464 = log 264/log 24
= 6/2 =3
Note: The examples have been considered with integers, all for ease of understanding and calculation. Logarithm of a number can be a decimal as well.