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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 am = 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 am = b           (Exponent form)

Then, m = logab      (Logarithm form)

 

Example
Express in logarithmic form: (i) 23 (ii) xm = y
Solution
Description: 10924.png
x m = y Description: 10918.png
 

 

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

  • Description: 12748.png
     
    Logarithm of a product of two numbers can be written as sum of logarithms of the individual numbers.
     
    Example
    If log102 = 0.3010, log103 = 0.4771, find log106.
    Solution
    log10 (6) = log10 (2 × 3)
     
    = log10 2 + log10 3
     
    = 0.3010 + 0.4771 = 0.7781
     
    Example
    Simplify log 2 + log 3 + log 4
    Solution
    log 2 + log 3 + log 4 = log (2 × 3 × 4) = log 24
     
  • Description: 10912.png 
     
    Logarithm of a division of two numbers can be written as difference of logarithms of the individual numbers.
     

     

    Example:
    Description: 10905.png.

     

    Example:

     

  • Description: 12815.png
     
    Logarithm of the nth power of a number is n times the logarithm of that number.
     
    Logarithm of a number m to the base a raised to power n is equal to 1/n times the logarithm of m to the base a.
     
     
    Example
    If log102 = 0.3010, find log10 8.
    Solution
    log10 8 = log10 23 = 3 log10 2 = 3 × 0.3010 = 0.9030
     
    Example
    Find the value of Description: 12825.png
    Solution
    Description: 12831.png
     
    Description: 12837.png
     
  • Description: 12865.png 
    Example
    Find the value of Description: 12877.png
    Solution
    Description: 12899.png
  • Description: 12908.png      [Change of base] 
     
    We can write any logarithm as a division of two logarithms (of the number and the base) taken at any common base. This theorem is very useful in solving problems having logarithms with many different bases.
     
    If we put m = a in the above result, we get another important result which is logna × logan = 1
     
    Example
    Find the value of Description: 13011.png
    Solution
    To simplify, we will make use of change of base theorem and convert each term into a logarithm with common base of 10.
     
    Description: 13034.png
  • If loga m = x, then
  1. log(1/a) m = -x
  2. Description: 13166.png 
  3. Description: 13160.png 





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