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 a 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 functions are written as

x = log₂ 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.

The formula for Change of Base

The logarithm is 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 log₁₀ by using the change of base formula.

Properties of Log Base 2

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

• • Product Rule : log₂ MN = log₂ M + log₂ N

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

Example : log 30 + log 2 = log 60

• • Quotient Rule : log₂ M/N = log₂ M – log₂ N

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

Example : log₂ 56 – log₂ 7 = log₂(56/7)=log₂8

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

Log₂ M^p = P log₂ M

• Zero Exponent Rule : log_a 1 = 0.
• Change of Base Rule : log_b (x) = ln x / ln b or log_b (x) = log₁₀ x / log₁₀ b
• Log_b b = 1 Example : log₂2 = 1
• Log_b b^x = x Example : log₂2^x = x

Sample Example

Question 1 :

Find the value of log₂ 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 log₂ 36 is 5.1699.

Question 2 :

Find the value of log₂ 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 log₂ 64 is 6.

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