If a is a positive real number, other than 1 and am = X, then we write:
m = loga x and we say that the value of log x to the base a is m.
Example:
(i) 103 = 1000 => log_{10}1000 = 3
(ii) 2^{-3} = 1/8 => log_{2} 1/8 = - 3
(iii) 3^{4} = 81 => log_{3}81=4
(iv) (.1)^{2} = .01 => log_{(.l)} .01 = 2
II. Properties of Logarithms:
1. loga(xy) = log_{a}x + loga y
2. log_{a}(x/y) = loga x - log_{a}y
3.log_{x}x=1
4. log_{a}1 = 0
5.log_{a}(x^{p})=p(log_{a}x)
6. log_{a}x =1/log_{x}a
7. log_{a}x = log_{b}x/log_{b}a=log x/log a.
Remember:
When base is not mentioned, it is taken as 10.
II. Common Logarithms:
Logarithms to the base 10 are known as common logarithms.
III. The logarithm of a number contains two parts, namely characteristic and mantissa.
Characteristic:
The integral part of the logarithm of a number is called its characteristic.
Case I: When the number is greater than 1.
In this case, the characteristic is one less than the number of digits in the left of the
decimal point in the given number.
Case II: When the number is less than 1.
In this case, the characteristic is one more than the number of zeros between the decimal
point and the first significant digit of the number and it is negative.
Instead of - 1, - 2, etc. we write, 1 (one bar), 2 (two bar), etc.
Mantissa:
The decimal part of the logarithm of a number is known is its mantissa