What is a binary logarithm?
The binary logarithm answers this question: “To what power must 2 be raised to get a number x?” It is written as log2(x). For example, log2(8) = 3 because 23 = 8.
Binary logarithms are especially common in computer science, information theory, and digital systems because modern computers operate in base 2 (bits).
Binary logarithm formula
If y = log2(x), then the equivalent exponential form is:
2y = x
For values that are not exact powers of 2, the result is a decimal approximation. Example: log2(10) ≈ 3.321928.
How to use this calculator
- Enter any positive real number in the input box.
- Click Calculate log2(x) (or press Enter).
- Read the result shown below the button.
- If the result is an integer, your number is an exact power of 2.
Examples
Exact powers of two
- log2(1) = 0
- log2(2) = 1
- log2(16) = 4
- log2(1024) = 10
Fractional and non-power values
- log2(0.5) = -1
- log2(3) ≈ 1.584963
- log2(10) ≈ 3.321928
Why binary logarithms matter
Binary logarithms show up in many practical settings:
- Algorithm complexity: Binary search runs in O(log2 n).
- Data structures: Heights of balanced trees scale with log2(n).
- Storage and memory: Address spaces and capacities are often powers of 2.
- Information theory: Entropy and bit-based measures use base-2 logs.
Domain and common mistakes
The binary logarithm is defined only for x > 0. You cannot take log2 of:
- Zero (undefined)
- Negative numbers (not real-valued in basic arithmetic)
If your input is invalid, the calculator displays a clear error message so you can correct it quickly.
Quick FAQ
Is log2(x) the same as ln(x)?
No. ln(x) is the natural logarithm (base e), while log2(x) uses base 2.
Can I enter decimals and scientific notation?
Yes. Inputs like 0.125, 12.75, and 1e9 are supported.
What does a negative answer mean?
A negative result means your input is between 0 and 1. Example: log2(0.25) = -2.