Log Base 2 Calculator — log₂(x)
Enter any positive number to get its base-2 logarithm (log₂), plus the natural log and log₁₀ for comparison — with a plotted curve showing where your value sits on the log₂ function.
Base-2 logarithm — the power to which 2 must be raised to equal x
- 1
ln(x)
ln(8) = 2.079442 - 2
ln(2)
0.693147Natural log of the base (2). - 3
log₂(x) = ln(x) ÷ ln(2)
2.079442 ÷ 0.693147 = 3
How does this calculator work?
log₂(x) = ln(x)/ln(2) = the power to which 2 must be raised to give x. It equals 0 at x=1, is negative for 0 < x < 1, and grows (slowly) for x > 1. Widely used in computing for bit counts, binary search depth and information theory (Shannon entropy in bits).
Formula
How this is calculated
The logarithm base 2 answers the question: "to what power must 2 be raised to give x?" So log₂(8) = 3 because 2³ = 8, and log₂(0.5) = −1 because 2⁻¹ = 0.5. The base-2 logarithm is also called the binary logarithm (lb) and appears wherever base-2 arithmetic matters: computer science (bit counts, the depth of a binary tree, the number of comparisons in binary search), information theory (Shannon entropy in bits), and music (equal-temperament octave ratios).
All logarithms share the same identities regardless of base: log(ab) = log(a) + log(b), log(a/b) = log(a) − log(b), and log(aⁿ) = n · log(a). Changing between bases uses the change-of-base formula: log₂(x) = ln(x)/ln(2) = log₁₀(x)/log₁₀(2), where ln(2) ≈ 0.6931 and log₁₀(2) ≈ 0.3010.
The function is defined only for x > 0. It equals zero at x = 1, is negative for 0 < x < 1, and grows without bound — though increasingly slowly — as x increases. The curve plotted below shows this concave-down shape; your entered value is marked with a point.
Frequently asked questions
Binary search on a sorted list of n items takes at most ⌈log₂ n⌉ comparisons. Binary encoding uses ⌈log₂ n⌉ bits to represent n distinct values. Hash-table load factors, tree heights and sorting algorithm complexities (O(log n)) all use base-2 logarithms. Shannon entropy is also measured in bits, computed with log₂.
Use the change-of-base formula: logₐ(x) = logᵦ(x) / logᵦ(a). In practice: log₂(x) = ln(x)/0.6931 = log₁₀(x)/0.3010. Most calculators provide only ln and log₁₀ keys; divide either by the log of 2 in that base to get log₂.
The real-valued logarithm is undefined for zero and negative numbers. As x approaches 0 from the positive side, log₂(x) decreases without bound toward −∞. For complex-number extensions of the logarithm, the imaginary part encodes the argument (angle), but that is beyond everyday use.
Also known as
TG we-Calculate Editorial Team. (2026). Log Base 2 Calculator — log₂(x) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/log-2-calculator
TG we-Calculate Editorial Team. "Log Base 2 Calculator — log₂(x)." TG we-Calculate. 2026. https://we-calculate.com/calculator/log-2-calculator.
TG we-Calculate Editorial Team, "Log Base 2 Calculator — log₂(x)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/log-2-calculator
@misc{wecalculate_log_2_calculator, title = {Log Base 2 Calculator — log₂(x)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/log-2-calculator}}, year = {2026}, note = {TG we-Calculate} }
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