A logarithmic system which reveals the number of multiplying factors when a number is expressed as multiplying factors on the basis of $10$, is called common logarithm.

A British Mathematician, Henry Briggs studied the system of Neper’s logarithm (Natural Logarithm) comprehensively but interested to use $10$ as base instead of Napier’s constant in logarithmic system. Thus, the system of common logarithm was introduced and it become most useful than Natural Logarithm. Hence, Common logarithm is called as Brigg’s logarithm or Briggsian logarithm.

$10$ is the minimum number that is used in decimal system in denominator of a fraction. So, it is called as decimal logarithm. The number $10$ represents a Decade, therefore it is also called as Decadic logarithm.

In common logarithms, $10$ is used as base of logarithm. It is neither equal to $1$ nor belongs to negative real numbers. However, it belongs to positive real numbers. It can be written in mathematical form as follows.

$10\ne 1$ and $\notin {R}^{\u2013}$ but $\in {R}^{+}$.

These conditions made number $10$ is an eligible element to use it as base for common logarithms.

The system of common logarithm can be written as a mathematical form expression by using the concept of algebra. Assume, $y$ is an independent algebraic variable, belongs to real numbers $(y\in R)$.

Assume, the number $10$ and algebraic variable $y$ both are involved in exponential form relation by working $10$ as base and $y$ as power. Assume, the value of $10$ to the power $y$ is equal to another algebraic variable $x$. The relation between them can be written in mathematical form as given here.

${10}^{y}=x$

The logarithm of $x$ to base $10$ equals to $y$ as per definition of the logarithmic system. It is written as given here.

$y={log}_{10}x$

Common logarithmic system is a commonly used logarithmic system in mathematics. So, logarithm of $x$ to $10$ is simply written as logarithm of $x$

$y={log}_{10}x=logx$

The mathematical relation between exponential form expression and common logarithmic system is expressed as given here.

${10}^{y}=x\iff {log}_{10}x=y$

${10}^{y}=x\iff logx=y$

You have to understand that the logarithmic system is a common logarithmic system if logarithmic is displayed either with or without base $10$ in mathematics.

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