# Log Base 2

Log base 2, also known as binary logarithm which is the inverse function of the power of two functions. The general logarithm states that for every real number N, can be expressed in exponential form as

N = a^{x}

Here, ‘ a ‘ is a positive real number called as base and ‘ x ‘ is an exponent , the logarithm form is written as

Log_{a }N = x

Log base 2 is the power to which the number 2 must be raised to obtain the value of n. For any real number x, log base 2 function is written as

x = log_{2} n

Which is equal to

2^{x} = n

Note that the logarithm of base 0 does not exist and logarithms of negative values are not defined in the real number system.

## Formula for Change of Base

The logarithm are in the form of log base 10 or log base e or any other bases. Here is a formula to calculate logarithms to base 2 or log base 2. The formula is stated by

\(\log _{2}x=\frac{\log_{10}x}{\log_{10}2}\)

Since the general formula for change of base is given by

\(\log _{a}x=\frac{\log_{b}x}{\log_{b}a}\)

To find the value of log base 2, first convert it into common logarithmic functions,i.e log base 10 or log_{10} by using the change of base formula.

## Properties of Log Base 2

Some of the logarithmic function properties with base 2 is given as follows :

**Product Rule**: log_{2 }MN = log_{2}M + log_{2 }N

Multiply two numbers with base 2, then add the exponents.

Example : log 30 + log 2 = log 60

**Quotient Rule :**log_{2 }M/N = log_{2}M – log_{2 }N

Divide two numbers with the base 2, subtract the exponents.

Example : log_{2} 56 – log_{2} 7 = log_{2}(56/7)=log_{2}8

**Power Rule :**Raise an exponential expression to a power and multiply the exponents.

Log_{2 }M^{p} = P log_{2} M

**Zero Exponent Rule :**log_{a}1 = 0.**Change of Base Rule :**log_{b }(x) = ln x / ln b or log_{b }(x) = log_{10}x / log_{10}b- Log
_{b }b = 1 Example : log_{2}2 = 1 - Log
_{b }b^{x }= x Example : log_{2}2^{x}= x

## Sample Example

### Question 1 :

Find the value of log_{2 }36

### Solution :

Given x=36

Using the change of base formula,

= \(\log _{2}x=\frac{\log_{10}x}{\log_{10}2}\)

= \(\log _{2}36=\frac{\log_{10}36}{\log_{10}2}\)

= 1.556303 / 0.301030

= 5.1699 ( corrected to 4 decimal points )

Therefore, the value of log_{2 }36 is 5.1699.

### Question 2 :

Find the value of log_{2 }64

### Solution :

Given x=64

Using the change of base formula,

= \(\log _{2}x=\frac{\log_{10}x}{\log_{10}2}\)

= \(\log _{2}64=\frac{\log_{10}64}{\log_{10}2}\)

= 1.806180 / 0.301030

= 6

Therefore, the value of log_{2 }36 is 6.