Negative Log Calculator — −log(x) in Any Base
The negative logarithm −log_b(x) flips the sign of the standard logarithm, making small values produce large positive results. It is the foundation of pH (−log₁₀[H⁺]), pKa, pOH, and similar "p-scale" quantities in chemistry and biochemistry. Enter any positive number and pick your base to compute it instantly.
Logarithm base
Negative logarithm of the input value
- 1
ln(x)
ln(0.001) = -6.907755 - 2
ln(base)
ln(10) = 2.302585 - 3
log_b(x) by change of base
-6.907755 ÷ 2.302585 = -3log_b(x) = ln(x) ÷ ln(b). - 4
−log_b(x)
−(-3) = 3
How does this calculator work?
−log_b(x) = −ln(x)/ln(b). For x between 0 and 1 the result is always positive. Base 10 is used for pH (= −log₁₀[H⁺]), pKa, and decibels; base e for thermodynamics and kinetics; base 2 for information-theoretic entropy. The result equals zero when x = 1.
Formula
How this is calculated
The logarithm log_b(x) answers "to what power must I raise b to get x?" The negative logarithm simply reverses the sign, so −log_b(x) = −ln(x)/ln(b). A key property: because log of a number less than 1 is negative, the negative log of a number between 0 and 1 is positive — which is why pH values are positive for acidic solutions where [H⁺] < 1 mol/L.
For base 10, the result shifts by 1 for every order-of-magnitude change in x: −log₁₀(0.001) = 3, −log₁₀(0.0001) = 4. For the natural log base e (≈ 2.71828), the shift per order of magnitude is larger (≈ 2.303). Base 2 is used in information theory where −log₂(p) gives the number of bits needed to encode an event with probability p.
The function is defined only for x > 0 and the base must be positive and not equal to 1. The result is positive when x < 1, zero when x = 1, and negative when x > 1 (for bases > 1). The XY plot shows how −log_b changes with x across a range centred on your input, with the input point highlighted.
Frequently asked questions
pH = −log₁₀([H⁺]), where [H⁺] is the molar concentration of hydrogen ions. Pure water has [H⁺] = 10⁻⁷ mol/L, so pH = 7. An acid with [H⁺] = 10⁻³ mol/L has pH 3. Enter the concentration (e.g. 0.001) in the x field with base 10 to compute the pH.
Both compute the negative logarithm but use different bases. Base-10 (common log) is used in pH, pKa, and decibel calculations. The natural log (base e ≈ 2.71828) appears in thermodynamics, reaction kinetics, and information theory. For the same x, −ln(x) ≈ 2.303 × (−log₁₀(x)).
Yes — when x > 1. For bases greater than 1, log_b(x) > 0 when x > 1, so −log_b(x) < 0. For example, −log₁₀(100) = −2. In chemistry this would correspond to a "negative pH", which is physically possible in extremely concentrated strong acids (superacids).
Also known as
TG we-Calculate Editorial Team. (2026). Negative Log Calculator — −log(x) in Any Base [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/negative-log-calculator
TG we-Calculate Editorial Team. "Negative Log Calculator — −log(x) in Any Base." TG we-Calculate. 2026. https://we-calculate.com/calculator/negative-log-calculator.
TG we-Calculate Editorial Team, "Negative Log Calculator — −log(x) in Any Base," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/negative-log-calculator
@misc{wecalculate_negative_log_calculator, title = {Negative Log Calculator — −log(x) in Any Base}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/negative-log-calculator}}, year = {2026}, note = {TG we-Calculate} }
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