Exponent Calculator · Practical guide
Log Base 2 Calculator
log2(x) answers "2 to what power equals x?" — log2(1024) = 10 and log2(256) = 8.
At a glance
What Log Base 2 Means
The base-2 logarithm answers the question: 2 raised to what power gives me x? Since 2^10 = 1,024, we have log2(1024) = 10. For numbers between powers of two the result is a decimal: log2(1,000) ≈ 9.965784, because 2^9.965784 ≈ 1,000.
Interactive tool
Exponent Calculator
Enter your own values, then compare the result with the examples below.
Results
Common log2 Values
| x | log2(x) | x | log2(x) |
|---|---|---|---|
| 2 | 1 | 64 | 6 |
| 4 | 2 | 128 | 7 |
| 8 | 3 | 256 | 8 |
| 16 | 4 | 512 | 9 |
| 32 | 5 | 1,024 | 10 |
The Change of Base Formula
Most calculators only have ln and log10 buttons, so base-2 logs are computed with log2(x) = ln(x) / ln(2). For example, ln(256) / ln(2) = 5.545177 / 0.693147 = 8. The calculator above applies this automatically when you select the logarithm operation with base 2.
Why Computer Science Runs on log2
- Bits: representing N distinct values needs log2(N) bits, rounded up — 1,024 values need exactly 10 bits.
- Binary search: finding an item in a sorted list of 1 million entries takes at most log2(1,000,000) ≈ 20 comparisons.
- Balanced trees: the depth of a full binary tree with 1,024 leaves is 10 levels.
- Data sizes: each doubling of memory adds one to the log — 1 KB = 2^10 bytes, 1 MB = 2^20.
Frequently Asked Questions
What is log2 of 1?
Zero. Any logarithm of 1 equals 0 because any base to the power 0 is 1, including 2^0 = 1.
Can log base 2 give a negative result?
Yes, for inputs between 0 and 1. log2(0.5) = -1 because 2^-1 = 0.5, and log2(0.25) = -2.
How do I calculate log2 without a log2 button?
Use the change of base formula: divide the natural log (or base-10 log) of your number by the natural log (or base-10 log) of 2. log2(100) = log(100) / log(2) ≈ 6.643856.
Why is log2 undefined for zero and negatives?
No power of 2 ever reaches 0 or a negative number, so the logarithm has no real answer there. The domain of log2 is strictly positive numbers.