3.0 Relation between common logarithm $\left( {{{\log }_{10}}x} \right)$ and Natural logarithm $\left( {{{\log }_e}x} \right)$

1.1 Case 1: When $a=1$
1.2 Case 2: When $a=0$
1.3 Case 3: When $N<0$

3.0 Relation between common logarithm $\left( {{{\log }_{10}}x} \right)$ and Natural logarithm $\left( {{{\log }_e}x} \right)$
2.1 Proofs of all the above properties
2.2 System of logarithms

The relaition between common logarithm $\left( {{{\log }_{10}}x} \right)$ and natural logarithm $\left( {{{\log }_e}x} \right)$ is given by,

$$\begin{equation} \begin{aligned} {\log _{10}}x = \frac{{{{\log }_e}x}}{{{{\log }_e}10}} \\ \frac{1}{{{{\log }_e}10}} = 0.434 \\ {\log _{10}}x = 0.434 \times {\log _e}x \\\end{aligned} \end{equation} $$ The factor $\left( {\frac{1}{{{{\log }_e}10}}} \right)$ is known as modulus of the common system.

When the factor $\left( {\frac{1}{{{{\log }_e}10}}} \right)$ is multiplied with natural logarithm or Napierian logarithm, it converts into common logarithm.