Subtraction of logarithms

Law of subtraction of logarithms:

"When two logarithms (of the same base) are subtracted, the result is a logarithm of the quotient of the two logarithms"
Note: The above law applies to logarithms with same bases, not with different bases.


  • `log_10(40)-log_10(2)=log_10(50/2)=log_10(25)`
  • `log_m(x)-log_m(y)=log_m(x/y)`
  • `log(x^2y^2)-log(xy)=log((x^2y^2)/(xy))=log(xy)`
  • `log(6x)-log(9x)=log((6x)/(9x))=log(1/3)`

Subtracting logarithms with different bases:

Logarithms with different bases can not be subtracted by the above mentioned law. However, by converting their bases to a common base, the logarithms can then be simplified as shown in the following example: 
`=  {log2 * log(30) - log(3) * log(20)}/{log(20) * log(30)}`

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