Calculate the value of **log _{2}(x)** for any (

**positive**) x with your chosen tool. When the log’s base is equal to 2, the operation is a specific instance of the logarithm. As a result, it was sometimes referred to as the binary logarithm. To get the log base 2 of any integer, enter it into the calculator below. Then, simply insert that value into the calculator below to get the log in any other base.

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## What is a logarithm?

The logarithm is the inverse function of exponentiation in **mathematics**. One logarithm of a given number x is the exponent to which another **fixed number**. The base b, must be increased to obtain that number x. The logarithm counts the number of times the same factor appears in repeated **multiplication **in its most basic form.

The exponent or power to which a base must be increased to generate a particular number is called a logarithm. If **b _{x} =**

**n**, then x is the logarithm of n to the base b, which is written as

**x = log**n in mathematics. For example,

_{b}**2**, therefore 3 is the base 2 logarithm of 8, or

^{3}= 8**3 = log**. In the same way, since

_{2}8**10**

^{2}equals

**100, 2**equals

**log**. The second type of logarithm (that is, logarithms of base 10) is referred to as common, or Briggsian, logarithms and is represented simply as log n.

_{10}100Logarithms, which were invented in the** 17th century** to speed up calculations, greatly lowered the time necessary to **multiply integers **with **multiple digits**. They were essential in numerical work for almost 300 years, until the development of mechanical calculating machines in the late 1800s and computers in the 1900s rendered them obsolete for large-scale calculations.

## Binary logarithm

The **binary logarithm** (**log _{2} n**) is the power to which the number 2 must be increased to get the value n in mathematics. For every real number x, that is. The binary logarithm of one is 0, the binary logarithm of two is one, the binary logarithm of four is two, and the binary logarithm of 32 is five.

The binary logarithm is the inverse of the power of two functions and is the logarithm to base 2. In addition to log_{2}, lb is another notation for a binary logarithm.

The binary logarithm of a **frequency ratio **of two musical tones gives the number of octaves by which the **tones differ**. The first application of binary logarithms was in music theory by Leonhard Euler: the binary logarithm of a frequency ratio of two musical tones gives the number of octaves by which the tones differ. Also, in information theory, binary logarithms can be used to compute the length of a number’s representation in the binary numeral system or the number of bits required to encode a message. They count the number of steps required for binary search and similar algorithms in computer science. Combinatorics, bioinformatics, sports tournament design, and photography are among domains where the binary logarithm is often utilized.

## Derivative of log base 2

The **log _{a}** x derivative (log x with base a) is

**1/ (x ln a)**. The fact that “ln” appears in the derivative of “log x” is intriguing. It’s worth noting that “ln” stands for natural logarithm (or logarithm with base “e”).

**ln = log**for example. In addition, the derivative of log x is

_{e},**1/(x ln 10)**because log’s default base is 10 if no base is specified.

The derivative of the binary logarithm function is:

(log_{2}x)’= 1/x 1/ln2 ≈ 1.44270/x

## How to Calculate Log Base 2?

According to the log rule,

Log Rule –

logb(x)=y

b^{y}=x

Example:

log_{2}32 = x

Using the log rule,

2^{x}= 32

We know that 32 in powers of 2 can be written 2^{5} or (2*2*2*2*2)

Therefore, x is equal to 5.