A logarithmic system which reveals the number of multiplying factors when a number is expressed as multiplying factors on the basis of $2$, is called binary logarithm.

Michael Stifel (or Styfel) was a German mathematician who introduced the system of a logarithms by taking $2$ as its base. The original meaning of binary is two pieces or two parts. Due to the usage of $2$ as base in logarithmic system, the logarithmic system is known as Binary logarithmic system.

The value of base $2$ is neither equal to $1$ nor belongs to negative real numbers group but it belongs to positive real numbers. These can be expressed in mathematical form.

$2\ne 1$ and $\notin {R}^{\u2013}$ but $\in {R}^{+}$.

The conditions made number $2$ is eligible to use as base in logarithm system as per the principle definition of the logarithms.

Binary logarithmic system can be written in mathematical form as an expression by applying the concept of algebra. Assume, $y$ is an algebraic variable, and it belongs to real numbers $(y\in R)$.

Assume, the number $2$ and the independent algebraic variable $y$ are involved in forming an exponential form expression by becoming base and power respectively. Also, assume the value of $2$ to the power $y$ is equal to another variable $x$. It can be expressed in mathematical form.

${2}^{y}=x$

According to definition of the logarithmic system, the logarithm of $x$ to base $2$ equals to $y$. It is written as given here.

$y={log}_{2}x$

The relation between exponential form expression and binary logarithmic system can be expressed in mathematical form.

${2}^{y}=x\iff {log}_{2}x=y$

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