Log Base 2: Formula, Rules and Examples

Log base 2 is one of the most common logarithms in math and computer science. It shows how many times you have to multiply 2 by itself to get that number. If you know how logarithms and exponents are related , then it is much easier to answer questions about log base 2 . In this guide you will learn what is log base 2, how to calculate it, convert to exponential form, use its properties and solve examples step by step.

Table of Contents

• What Is Log Base 2?
• Formula for Log Base 2
• How to Calculate Log Base 2
• Log Base 2 Value Table
• Properties of Log Base 2
• Log Base 2 in Computer Science
• Solved Examples of Log Base 2
• Frequently Asked Questions of Log Base 2

What Is Log Base 2?

Log base 2 of a number N is the power to which 2 must be raised to get N. If we write this as an equation, it looks like this:

log₂N = k.

Example: log₂32 = 5
Since 2 x 2 x 2 x 2 x 2 = 32, that is 2 raised to the power 5.

Formula for Log Base 2 

The formula for log base 2 expresses the relationship between logarithms and exponents. If a number N is equal to 2 raised to the power k, then the logarithm of N to base 2 is k.

Logarithmic form: log₂N = k   is exactly the same as Exponential form: 2ᵏ = N

How to Calculate Log Base 2

Case 1: The Number Is an Exact Power of 2

• Write the number as 2 raised to some power.

• That power is your answer.

Example: Find log base 2 of 512

Break 512 down as repeated doubling: 2, 4, 8, 16, 32, 64, 128, 256, 512. That is nine doublings starting from 2 itself, so 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁹ = 2 raised to the power 9.

Therefore, log₂512 = 9.

Case 2: The Number Is Not an Exact Power of 2

Numbers like 20, 50 or 100 cannot be written as a whole number power of 2, so we use the change of base formula instead, switching to a base your calculator already knows, such as base 10 or the natural log:

$\log_{2}(N)=\frac{\log(N)}{\log(2)}$

(using base 10 log or the same with natural log ln)

Example: Find log base 2 of 50 (approximately)

log₂50 = log 50 / log 2

Using a calculator, log 50 is approximately 1.699 and log 2 is approximately 0.301.

log₂50 = 1.699 / 0.301 ≈  5.64.

Since 2^5 is 32 and 2^6 is 64, 50 should give a value between 5 and 6.

You can also use: 

Log Base 2 Calculator

Log Base 2 Value Table

Let us look at some standard log base 2 values:

 

Properties of Log Base 2

Log base 2 follows the same fundamental rules as logarithms with any other base. Some important properties of log base 2 are given below.

• Log of 1

log₂1 = 0, because 2 raised to the power 0 equals 1.

• Log of the Base Itself

log₂2 = 1, because 2 raised to the power 1 equals 2.

• Product Rule

log₂a + log₂b = log₂(ab), so multiplication inside becomes addition outside.

• Quotient Rule

log₂a minus log₂b = log₂(a divided by b), so division inside becomes subtraction outside.

• Power Rule

log₂(a raised to the power n) = n times log₂ a, so an exponent inside moves out as a multiplying factor.

• Change of Base Rule

log₂N = log N divided by log 2, which lets you compute log base 2 using a standard base-10 or base-e calculator.

Log Base 2 in Computer Science

Log base 2 is one of the most commonly used calculations in computer science.

• Memory sizes: A computer memory chip labelled 1 GB actually stores 2^30 bytes, so log base 2 explains why memory sizes always look like 256, 512, 1024 and so on, instead of round numbers.

• Binary search: If a program has to search through a sorted list of a million items using binary search , the number of steps it needs is about log₂ (1000000). This is about 20 steps, a huge saving compared to checking every item one by one.

• Number of bits: If you want to know how many bits are needed to represent N different values, the answer is log₂N, rounded up to the next whole number.

Solved Examples of Log Base 2

Example 1: Find the value of log base 2 of 256.

Solution: Write 256 as a power of 2. Since 2^8 = 256, we get log2(256) = 8.

Example 2: Express  $9^k = 2^{18}$ using log base 2, and find k.

Solution: Given:  $9^k = 2^{18}$

Take log base 2 on both sides:

$log₂(9^k) = log₂(2^{18})$

Using the power rule: k log₂(9) = 18 log₂(2)

Since log₂(2) = 1 we get:

k log₂(9) = 18

Solve for k:

k = 18 / log₂(9)

Using the change of base formula:

k = 18 / (log(9) / log(2))

k ≈ 18 / 3.17

k ≈ 5.68

Therefore, k ≈ 5.68

Example 3: Simplify log₂64 - log₂4.

Solution: Using the quotient rule,  log₂64 - log₂4 = log₂ (64 / 4) = log₂16.

Since 16 =  $2^4$, the answer is 4.

Example 4: A memory chip stores  $2^{12}$ kilobytes. How many kilobytes is that, and what is log base 2 of that value?

Solution: $2^{12}$ = 4096, so the chip stores 4096 kilobytes.

Since the number is already written as a power of 2,  $log_24096 = 12$directly.