Power of 2 calculator

What is a power of 2 calculator?

A power of 2 calculator works with the expression $2^n$ — the number two multiplied by itself $n$ times. Powers of two appear everywhere in computing, because digital systems store and address information in binary, where each extra bit doubles the number of possible states.

This calculator runs in two directions. Given an exponent $n$, it returns the value $2^n$. Given a positive value $v$, it returns the exponent $n = \log_2 v$ and tells you whether $v$ is an exact power of two.

Key concepts

• Base — the number being raised to a power. Here the base is fixed at 2.
• Exponent (n) — how many times the base is multiplied by itself. It can be zero, negative, or non-integer.
• Power (2ⁿ) — the result of the exponentiation.
• Base-2 logarithm — the inverse operation: $\log_2 v$ answers “to what exponent must 2 be raised to get $v$?”.

How does the calculator work?

Use the Calculate selector to choose a direction. In “Value from exponent” mode you type $n$ and read off $2^n$. In “Exponent from value” mode you type a positive $v$ and read off $\log_2 v$, along with a note stating whether $v$ is an exact power of two.

Formulas

The value from an exponent:

The exponent from a value:

A value $v$ is an exact power of two when $\log_2 v$ is a whole number.

Worked examples

Example 1: value from exponent, n = 10

Example 2: value from exponent, n = 0

Example 3: value from exponent, n = 16

Example 4: exponent from value, v = 256

Because 8 is a whole number, 256 is an exact power of two: $2^8 = 256$.

Example 5: exponent from value, v = 100

Since 6.6439 is not a whole number, 100 is not an exact power of two — it falls between $2^6 = 64$ and $2^7 = 128$.

Practical uses

• Computing and memory — kilobytes, megabytes, and gigabytes are commonly defined in powers of two ($2^{10}$, $2^{20}$, $2^{30}$).
• Networking — subnet sizes and address ranges are powers of two; see the binary to decimal calculator for the underlying conversions.
• Algorithms — binary search, balanced trees, and divide-and-conquer methods scale with powers of two.
• Scientific notation — for very large or very small results, switch to the exponential notation calculator.

Notes

• The exponent may be negative: $2^{-3} = 0.125$.
• A non-integer exponent is allowed: $2^{0.5} = \sqrt{2} \approx 1.4142$.
• The base-2 logarithm is only defined for positive values, so $v$ must be greater than 0.
• A value is flagged as an exact power of two only when its base-2 logarithm rounds to a whole number within a tiny tolerance.