Log Base 2: Formula and Properties

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Log base 2 is used in writing exponential form with a base of 2 into logarithmic form. It is also known as binary logarithm. It must be noted that the logarithm of base 0 does not exist. Whereas logarithms of negative values are undefined in the real number system. The general logarithm signifies that for every real number n can be expressed in exponential form as:

n = ax

In the above equation, a is the positive real number known as base and x is an exponent. The logarithm form will be written as, Loga n=x

For every real number x, log base 2 functions will be written as, x = log₂n

Which is equal to, 2x = n

Read Here: Relations and Functions

Table of Contents

What is Log Base 2?
How to Find Log Base 2?
Log Change of Base Formula
Properties of Log Base 2
Things to Remember
Previous Year's Questions
Sample Questions

Key Terms: Log Base 2, Logarithm, Exponential Form, Functions, Number System

What is Log Base 2?

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Log Base 2 can be defined as a mathematical form denoting a natural number in an exponential form to the base of 2. For example, the exponential form of 24 = 16 can also be represented as, log216 = 4.

Log Base 2

Log N to the base of 2 is simply equivalent to denoting the number N in an exponential form with base 2. It can be represented as:

Logarithmic Form	Exponential Form
Log₂1 = 0	2⁰ = 1
Log₂2 = 1	2¹ = 2
Log₂4 = 2	2² = 4
Log₂8 = 3	2³ = 8
Log₂16 = 4	2⁴ = 16
Log₂32 = 5	2⁵ = 32
Log₂64 = 6	2⁶ = 64
Log₂128 = 7	2⁷ = 128
Log₂256 = 8	2⁸ = 256

How to Find Log Base 2?

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To find log base 2, we need to follow the rule below. According to the log rule:

log_b(x)=y

⇒ by = x

Suppose we have, log₂16 = x

Using the log rule,

⇒ 2x= 16

We know that 16 in powers of 2 can be written as (2×2×2×2 =16) ,2x=24

⇒ Therefore, x is equal to 4

Read Also:

Topic Related Concepts
Logarithm Formula	Value of Log Infinity	Value of Log 1 to 10

Log Change of Base Formula

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The logarithm can be in log base 10 or in the form of log base e or in any other bases. In order to know the value of log base 2, firstly it is required to convert it into log base 10 also called a common logarithm.

The formula to calculate logarithms to base 2 or log base 2 is:

\(log_2x = \frac{log_{10}x}{log_{10}2}\)

The general formula for change of base is given by:

\(log_ax = \frac{log_{b}x}{log_{b}a}\)

Properties of Log Base 2

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Below mentioned are a few of the logarithmic function properties with base

Product Rule: log₂ MN = log₂ M + log₂ N

Multiply two numbers with base 2, and add the exponents following that. Example: log 30 + log 2 = log 60.

Quotient Rule: log₂ M/N = log₂M – log₂N

Divide two numbers with the base 2, and following that, subtract the exponents. Example : log₂56 – log₂7 = log₂(56/7)=log₂8

Power Rule: Raise an exponential expression to power and multiply the exponents. Thus, Log₂ Mp = 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) = log₁₀x / log₁₀b

Log_b b = 1 ( an Example of this can be: log₂2 = 1)

Log_b bx = x, (an Example of this can be: log₂2x = x)

Also Read:

Chapter Related Topics
Types of Relations	Types of Functions	Binary Operations

Things to Remember

Log Base 2 is also called binary logarithm.
Logarithm of base 0 is non-existing and in the real number system, logarithms of negative values are undefined.
Log base 2 can be used to express exponential form with a base of 2 into logarithmic form.
The general formula for the change of base can be given as,

Read Also: Difference between ln and Log

Previous Year Questions

The differential coefficient of … [KCET 2004]
The differential coefficient of log10x … [KCET 2016]
If f(x)= logx {ln(x)}, then … [AMUEEE 2012]
The value of … [JKCET 2013]
ddxxx is equal to … [BITSAT 2011]

Sample Questions

Ques. What is the value of Log2128? (1 mark)

Ans. The value of Log2128 is 7.

Ques. What is the approximate value of log 2? (1 mark)

Ans. The approximate value of log 2 is 0.3

Ques. Determine the value of 1024 to the log base 2. (2 marks)

Ans. The number 1024 can be solved using log base 2:

Log21024 = log2210

= 10log22

= 10

Ques. What is the derivative of Log Base 2 to x? (2 marks)

Ans. The derivative of log base 2 to x is equivalent to 1/x.log2. Further, it can be expressed as:

d/dx.log2x

= d/dx.logx/log2

= 1/x . 1/log2

= 1/xlog2

Ques. What are some uses of Logarithms in our everyday life? (3 marks)

Ans. Some of the major uses of Logarithms in everyday life are:

Earthquakes and amplitude are recorded on seismographs and the Richter scale respectively. To understand the values, Logarithmic values are used.
Logarithmic values are also used to determine the pH value of substances.
Logarithms are used to measure sound intensity.
Logarithms are also used to calculate complex values.

Ques. Determine the value of log₂36. (3 marks)

Ans. Given x = 36

Now, after applying the change of base formula,

=log₂x = log₁₀xlog₁₀2

= log₂36 = log₁₀36/log₁₀2

= 1.556303 / 0.301030

= 5.1699 (corrected to 4 decimal points)

Hence, the value of log236 is 5.1699.

Ques. What is the value of log 2 base 10? (3 marks)

Ans. The value of log 2 base 10 can be calculated by the rule,

Log_a(b) = log_b/log_a

Log₁₀(2) = log₂/log₁₀ = 0.3010

Hence, the value of log 2 base 10 = 0.3010.

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