Log base 2

• A Logarithmic Function is a function that is the inverse of an exponential function.

• The purpose of the logarithm is to tell us about the exponent.

• Logarithmic Functions are used to explore the properties of exponential functions and are used to solve various exponential equations.

Log base 2 is an inverse representation of the power of 2. For example, n = bx  here, n is a real positive number. And x is the exponent number. Then, the log base format of this is Logb n = x. 

What is Log Base 2 or Binary Logarithm?

• Log base 2 is also known as binary logarithm.

• It is denoted as (log2n).

• Log base 2 or binary logarithm is the logarithm to the base 2.

• It is the inverse function for the power of two functions.

• Binary logarithm is the power to which the number 2 must be raised in order to obtain the value of n.

• Here’s the general form.

\[{\mathbf{x}}{\text{ }} = {\text{ }}{\mathbf{lo}}{{\mathbf{g}}_{\mathbf{2}}}{\mathbf{n}}\;\;\;\boxed{} -  -  -  -  -  -  -  -  - \boxed{}{{\mathbf{2}}^{\mathbf{x}}} = {\text{ }}{\mathbf{n}}\]

Graph for Log Base 2

Properties of Log Base 2

There are a few properties of logarithm functions with base 2. They are listed in the table below.

Since we are discussing log base2, we will consider the base to be 2 here.

Basic Log Rules

Here are a Few Examples That show How the above Basic Rules work

Example 1 –

Log 40, which can be further written as,

Log (20× 2)

by-product rule 

= log 20 + log 2

which is equal to log 40

Example 2 – 

Find the value of log4(4)?

 logb(b) = 1, by identity rule

Therefore, log4(4) = 1.

The Formula for Change of Base

The logarithm can be in the form of log base e or log base 10 or any other bases. Here’s the general formula for change of base -

\[{\mathbf{Lo}}{{\mathbf{g}}_{\mathbf{b}}}\;{\mathbf{x}}{\text{ }} = {\text{ }}\frac{{{\mathbf{Lo}}{{\mathbf{g}}_{\mathbf{a}}}\;{\mathbf{x}}}}{{{\mathbf{Lo}}{{\mathbf{g}}_{\mathbf{a}}}\;{\mathbf{b}}}}\;\]

To find the value of log base 2, we first need to convert it into log base 10 which is also known as a common logarithm.

\[Log{\text{ }}base{\text{ }}2{\text{ }}of{\text{ }}x = \frac{{ln\left( x \right)}}{{ln\left( 2 \right)}}\]

Now You might be wondering What Common Logarithmic Function is?

• Common Logarithmic Function or Common logarithm is the logarithm with base equal to 10.

• It is also known as the decimal logarithm because of its base.

• The common logarithm of x is denoted as log x.

• Example: log 100 = 2 (Since 102= 100).

How to Calculate Log Base 2?

This is how to find log base 2 -

• According to the log rule,

Log Rule - 

\[log_{b}(x) = y\]

\[b^{y} = x\]

•  Suppose we have a question, 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.

Questions to be Solved –

Question 1) Calculate the value of log base 2 of 64.

Solution) Here, 

X= 64

Using the formula,

\[Log{\text{ }}base{\text{ }}2{\text{ }}of{\text{ }}X = \frac{{ln\left( {64} \right)}}{{ln\left( 2 \right)}} = 6\].

Log base 2 of 64 =\[\frac{{ln\left( {64} \right)}}{{ln\left( 2 \right)}} = 6\].

Therefore, Log base 2 of 64 = 6

Question 2) Find the value of log2(2).

Solution) To find the value of log2(2) we will use the basic identity rule,

\[lo{g_b}\left( b \right){\text{ }} = {\text{ }}1,\].

Therefore, log2(2) = 1.

Question 3) What is the value of log 2 base 10?

Solution) The value of log 2 base 10 can be calculated by the rule,

\[Lo{g_a}\left( b \right){\text{ }} = \frac{{\log b}}{{\log a}}\].

\[Lo{g_{10}}\left( 2 \right){\text{ }} = \frac{{\log 2}}{{\log 10}}\; = {\text{ }}0.3010\].

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

Question 4) What is the value of log 10 base 2?

Solution) The value of log 10 base 2 can be calculated by the rule,

\[Lo{g_b}\left( a \right){\text{ }} = \frac{{\log b}}{{\log a}}\].

\[Lo{g_2}\left( {10} \right){\text{ }} = \frac{{\log 10}}{{\log 2}}\; = {\text{ 3}}{\text{.3 = 2}}\].

Therefore, the value of log 10 base 2 = 3.32.

Uses of Logarithms in Everyday Life

• Earthquakes are recorded on seismographs and the amplitude is recorded on the Ritcher scale. Logarithmic values are used to comprehend these values

• It is also used in determining the pH value of any substance

• logarithms are used in measuring the sound intensity. Generally, sound intensity is measured by loudness which in turn is measured using logarithms

• They are also used in measuring complex values.

How to improve Scores in the Logarithm Chapter

Logarithm is just the opposite of expressing a number to the power of a digit. Many students face difficulty with this subject as they have to think and solve the problems in reverse. Following are some tips to improve your scores in logarithms:

• Understand that logarithm is an inverse expression of powers or exponents. All you have to do is solve them inversely

• Byheart learn all the laws of the logarithm and know what would be the end result for a particular problem if they solve it with the help of a formula

• Understanding the end result can be done by understanding the formula you apply to solve a problem. When you apply a formula that has the answer, you will get to know the end result

• Practice as many problems as you can with different logarithmic values

• Refer to the previous year questions to know the exam pattern, types of questions asked in the exam and assess the depth of the questions asked by the examiner.

• Any doubts can be clarified with your subject teacher or can be clarified from an online learning website like Vedantu.