# Log Base 2 — Binary Logarithm, Values & Examples

**TL;DR**

Log base 2 of a number is the power to which 2 must be raised to produce that number — so \( \log_2 8 = 3 \) because \( 2^3 = 8 \). This article covers what log base 2 (the binary logarithm) means, how to compute it by hand and with the change-of-base formula, a value table, its properties, where it powers computer science, and the slips to avoid.

## What Is Log Base 2?

**Log base 2** of a number \( x \), written \( \log_2 x \), is the exponent you put on 2 to get \( x \). In symbols:

\[ \log_2 x = y \Longleftrightarrow 2^y = x. \]

So \( \log_2 8 = 3 \) because \( 2^3 = 8 \), and \( \log_2 32 = 5 \) because \( 2^5 = 32 \). The logarithm and the exponent are the same fact looked at from opposite ends — one asks "what is \( 2^3 \)?", the other asks "2 to _what_ gives 8?". That two-way relationship is the whole of log to exponential form conversion.

Log base 2 is also called the **binary logarithm**, sometimes written \( 	ext{lb} \) or \( 	ext{lg} \). The "binary" name is a hint about where it earns its keep — anything built on two states, on/off, 0/1.

## How Do You Calculate Log Base 2?

There are two cases, and which one you are in decides the method.

**When \( x \) is a power of 2,** read the answer straight off. Ask how many times you multiply 2 to reach \( x \):

\[ \log_2 16 = 4 \quad \text{since} \quad 2^4 = 16. \]

No calculator needed. The powers of 2 — 1, 2, 4, 8, 16, 32, 64, 128, 256 — are worth knowing on sight.

**When \( x \) is not a power of 2,** use the **change-of-base formula**. It states:

\[ \log_2 x = \frac{\ln x}{\ln 2} = \frac{\log x}{\log 2}. \]

For example,

\[ \log_2 10 = \frac{\ln 10}{\ln 2} \approx \frac{2.3026}{0.6931} \approx 3.32. \]

## Log Base 2 Value Table

For the exact powers of 2, the binary logarithm is a clean whole number. This table is the one to memorise.

| \( x \) | \( \log_2 x \) | Because |
| --- | --- | --- |
| 1 | 0 | \( 2^0 = 1 \) |
| 2 | 1 | \( 2^1 = 2 \) |
| 4 | 2 | \( 2^2 = 4 \) |
| 8 | 3 | \( 2^3 = 8 \) |
| 16 | 4 | \( 2^4 = 16 \) |
| 32 | 5 | \( 2^5 = 32 \) |
| 64 | 6 | \( 2^6 = 64 \) |
| 128 | 7 | \( 2^7 = 128 \) |
| 256 | 8 | \( 2^8 = 256 \) |
| 1024 | 10 | \( 2^{10} = 1024 \) |

Two boundary facts close the table: \( \log_2 1 = 0 \) (anything to the power 0 is 1), and \( \log_2 0 \) is **undefined** — no power of 2 ever reaches 0.

## What Are the Properties of Log Base 2?

Log base 2 obeys the same laws as every logarithm — they just carry a subscript 2. These come straight from the logarithm rules, specialised to base 2.

- **Product rule:** \( \log_2(MN) = \log_2 M + \log_2 N \) — a product turns into a sum.
- **Quotient rule:** \( \log_2 \left( \frac{M}{N} \right) = \log_2 M - \log_2 N \) — a quotient turns into a difference.
- **Power rule:** \( \log_2(M^p) = p, \log_2 M \) — an exponent comes down front.
- **Base of itself:** \( \log_2 2 = 1 \), and more generally \( \log_2 2^k = k \).
- **Log of 1:** \( \log_2 1 = 0 \) in every base.

## Examples of Log Base 2

### Example 1

**Find \( \log_2 64 \).**

\[ \log_2 64 = 6 \] because \( 2^6 = 64. \)  
**Final answer:** 6.

### Example 2

**Find \( \log_2 12 \) — and watch the tempting wrong move first.**

**Wrong attempt:** \( \log_2 12 = 2 \times 3 = 6. \) Check it: \( 2^6 = 64 \), nowhere near 12. The answer cannot be 6.

**Correct:** \( \log_2 12 = \log_2(4 \times 3) = \log_2 4 + \log_2 3 = 2 + \log_2 3. \)

\( \log_2 3 \approx 1.585 \), so \( \log_2 12 \approx 3.585. \)  
**Final answer:** \( \log_2 12 \approx 3.585. \)

### Example 3

**Find \( \log_2 \frac{1}{8} \).**

\( 1/8 = 2^{-3} \), so \( \log_2 \frac{1}{8} = -3. \)  
**Final answer:** -3.

### Example 4

**Use the change-of-base formula to find \( \log_2 50 \).**

\( \log_2 50 = \frac{\ln 50}{\ln 2} \approx 5.64. \)  
**Final answer:** \( \log_2 50 \approx 5.64. \)

### Example 5

**Simplify \( \log_2 8 + \log_2 4 \) using the product rule.**

\[ \log_2 8 + \log_2 4 = \log_2(8 \times 4) = \log_2 32 = 5. \]  
**Final answer:** 5.

### Example 6

**How many bits are needed to store a number from 0 to 999?**

Storing \( N \) distinct values needs \( \lceil \log_2 N \rceil \) bits — the binary logarithm rounded up. With 1000 values:

\[ \log_2 1000 = \frac{\ln 1000}{\ln 2} \approx 9.97, \qquad \lceil 9.97 \rceil = 10. \]

**Final answer:** 10 bits.

## Why Log Base 2 Runs Through Computing

Log base 2 is the native tongue of computers, measuring counting bits and powering algorithms. It’s crucial in areas like:

- **Bits to represent a number.** A value up to N needs \( \lceil \log_2 N \rceil \) bits.
- **Algorithm speed.** Binary search, balanced search trees, and merge sort run in time proportional to \( \log_2 n \).
- **Information theory.** Claude Shannon measured information in bits with binary logarithm at its core.

## Where Students Trip Up on Log Base 2

Common mistakes include:

1. Forgetting the base and using base 10 by reflex.
2. Taking the log of zero or a negative number.
3. Multiplying logs instead of adding them.

## The Short Version

- Log base 2 of \( x \) is the power of 2 that gives \( x \).
- For powers of 2, read the answer off the value table; for other numbers, use the change of base.
- The product, quotient, and power rules apply with a subscript 2.
- The most common mistake is forgetting the base.
- Log base 2 measures information in bits and the step-count of halving algorithms.

## Practice These Before Moving On

1. Find \( \log_2 128 \).
2. Find \( \log_2 \frac{1}{16} \).
3. Use change of base to estimate \( \log_2 20 \).
4. Simplify \( \log_2 16 - \log_2 2 \).
5. How many bits store a number from 0 to 500?

**Answers:** 1. 7, 2. -4, 3. ≈ 4.32, 4. 3, 5. 9 bits.