Logarithm
The logarithm of a number to a given base is the index or the power to which the base must be raised to obtain that number.
If a^{m} = b (where b > 0, a ≠ 1, a > 0), then the exponent m is said to be logarithm of b to the base a.
So, if a^{m} = b (Exponent form)
Then, m = log_{a}b (Logarithm form)
Example
Express in logarithmic form: (i) 2^{3} (ii) x^{m} = y
Solution
x ^{m} = y
Note:
 Logarithm of any quantity to the same base is 1.
 Logarithm of 1 to any base is zero.
Types of Logarithm
Logarithms to the base “e” (exponential constant) are called natural logarithms.Logarithms to the base “10” are called common logarithms.
Note: If the base is not given, it can be assumed to be 10.
Laws of Logarithm

ExampleIf log_{10}2 = 0.3010, log_{10}3 = 0.4771, find log_{10}6.Solutionlog_{10 }(6) = log_{10} (2 × 3)ExampleSimplify log 2 + log 3 + log 4Solutionlog 2 + log 3 + log 4 = log (2 × 3 × 4) = log 24

Example:
.Example:

ExampleIf log_{10}2 = 0.3010, find log_{10} 8.Solutionlog_{10} 8 = log_{10} 2^{3} = 3 log_{10} 2 = 3 × 0.3010 = 0.9030ExampleFind the value ofSolution 
ExampleFind the value ofSolution
 [Change of base]
ExampleFind the value ofSolutionTo simplify, we will make use of change of base theorem and convert each term into a logarithm with common base of 10.
 If log_{a} m = x, then
 log_{(1/a)} m = x