# Binary Logarithm

## Definition

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 $\notin {R}^{–}$ but .

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

## Expression

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 $\left(y\in R\right)$.

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.

${2}^{y}=x⇔{log}_{2}x=y$

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