Logarithm: 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


  1. \(10^{3}-1000\)    => \(\log _{10} 1000=3\) 
  2. \(3^{4}=81 \quad \Rightarrow \log _{3} 81=4\)
  3. \(2^{-3}=\frac{1}{8} \quad=>\quad \log _{2} \frac{1}{8}=-3\)

Proportion of Logarithms

  • \(\log _{a}(x y)\)\(\log _{a} x+\log _{a} y\)
  • \(\log _{a}\left(\frac{x}{y}\right)=\log _{a} x-\log _{a} y\)
  • \(\log _{x} x=1\)
  • \(\log _{a} 1=0\)
  • \(\log _{a}\left(x^{p}\right)=p\left(\log _{a} x\right)\)
  • \(\log _{a} x=\frac{1}{\log _{x} a}\)
  • \(\log _{a} x\) =  \(\frac{\log _{b} x}{\log _{b} a}=\frac{\log x}{\log a}\)

Remember: When base is not mentioned, it is taken as 10 .

Common Logarithms

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 integral part of the logarithm of a number is called it is characteristic.

Case 1: 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 2: 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, \(\overline{1} \, (one bar)\), \(\overline{2}\) (two bar), ete.