Logarithms
If a is a positive real number, other than 1 and a^{m} = x, then we write:
m = log_{a}x and we say that the value of log x to the base a is m.
Examples:
(i). 10^{3} 1000 log_{10} 1000 = 3.
(ii). 3^{4} = 81 log_{3} 81 = 4.
(iii). 2^{3} =  1  log_{2}  1  = 3. 
8  8 
(iv). (.1)^{2} = .01 log_{(.1)} .01 = 2.
Properties of Logarithms:
1. log_{a} (xy) = log_{a} x + log_{a} y
2. log_{a}  x  = log_{a} x  log_{a} y  
y 
3. log_{x} x = 1
4. log_{a} 1 = 0
5. log_{a} (x^{n}) = n(log_{a} x)
6. log_{a} x =  1 
log_{x} a 
7. log_{a} x =  log_{b} x  =  log x  . 
log_{b} a  log a 
MIND IT !

Common Logarithms:
Logarithms to the base 10 are known as common logarithms.

The logarithm of a number contains two parts, namely 'characteristic' and 'mantissa'.
Characteristic: The internal 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.
Examples:
Number Characteristic Number Characteristic 654.24 2 0.6453 1 26.649 1 0.06134 2 8.3547 0 0.00123 3 Mantissa:
The decimal part of the logarithm of a number is known is its mantissa. For mantissa, we look through log table.