Negative Log Calculator

Enter a positive value and a base to compute the negative logarithm −logb(x), shown alongside the equivalent values in base 10, base e, and base 2.

Quick Facts

Definition
−log_b(x) = −ln(x) / ln(b)
Flips the sign of the ordinary logarithm; positive whenever 0 < x < 1.
pH scale
pH = −log₁₀[H⁺]
Turns a tiny hydrogen-ion concentration into a readable positive number.
Natural log
−ln(x)
Appears in decay, entropy, and likelihood formulas.
Binary log (bits)
−log₂(p)
Shannon information content, in bits, of an event with probability p.

Your Results

Calculated
Negative Log (selected base)
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−log_b(x) using your chosen base
Common Log (base 10)
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−log₁₀(x) — e.g. pH scale
Natural Log (base e)
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−ln(x)
Binary Log (base 2)
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−log₂(x) — e.g. bits of information

Ready

Enter a positive value and choose a base, then press Calculate.

How the Negative Log Calculator Works

The negative logarithm of a positive number x in base b is −logb(x) — the ordinary logarithm with its sign flipped. Because logb(x) is negative for any x between 0 and 1, negating it turns small fractions and probabilities into convenient positive numbers. That single trick underlies scales such as pH, statistical p-value plots, and Shannon information content. This calculator computes −logb(x) for the base you choose, plus the equivalent value in base 10, base e (natural log), and base 2 so you can compare all four at once.

Formula and change of base

For any positive x and any valid base b (b > 0, b ≠ 1): −logb(x) = −ln(x) / ln(b). This is the standard change-of-base identity applied to the ordinary logarithm and then negated. Because logarithms of the same number in different bases are just constant multiples of one another, converting between bases only requires dividing by ln(b) instead of ln(10) or ln(2). Note that logb(x) — and therefore −logb(x) — is defined only when x > 0; there is no real logarithm of zero or a negative number, and the base itself must be positive and not equal to 1.

Why the sign gets flipped

For 0 < x < 1, logb(x) is negative, so −logb(x) is positive — this is why pH, defined as pH = −log₁₀[H⁺], comes out as a small positive number even though hydrogen-ion concentrations [H⁺] are tiny fractions like 0.0000001 M. The same pattern appears in information theory, where the information content (in bits) of an outcome with probability p is −log₂(p): rarer outcomes (smaller p) carry more information (a larger −log₂(p)). Statisticians use −log₁₀(p-value) the same way, turning very small p-values into readable positive numbers for plots such as Manhattan plots in genetics.

Common mistakes

  • Entering zero or a negative x: the logarithm is undefined there — the input must be strictly greater than 0.
  • Using a base of 1 (or 0, or a negative base): log base 1 is undefined because 1 raised to any power is always 1.
  • Forgetting the sign: reporting logb(x) instead of −logb(x) (or vice versa) reverses the direction of the scale — for example, turning a "more acidic" pH reading into a "less acidic" one.
  • Mixing bases when comparing values: a −log₂ value and a −log₁₀ value computed from the same x are not directly comparable without converting them to a common base first.

Real-world applications

  • Chemistry: pH = −log₁₀[H⁺] and pOH = −log₁₀[OH⁻] describe how acidic or basic a solution is.
  • Statistics and genomics: −log₁₀(p) rescales tiny p-values so they can be plotted on a readable positive axis.
  • Information theory: −log₂(p) measures the information content, in bits, carried by an event of probability p (Shannon's formula), the basis of entropy calculations.
  • Signal and acoustic fields: negative logarithms convert attenuation and loss ratios into positive figures that are easier to tabulate and compare.

Frequently Asked Questions

What is a negative logarithm?
The negative logarithm of a positive number x in base b is −logb(x): the ordinary logarithm of x, multiplied by −1. Because logb(x) is negative whenever 0 < x < 1, its negative is positive — a convenient way to express very small numbers as manageable positive values, as in the pH scale.
How do I calculate −log_b(x) by hand?
Use the change-of-base formula: −logb(x) = −ln(x) / ln(b), where ln is the natural logarithm. For example, −log₁₀(0.001) = −ln(0.001)/ln(10) = −(−6.9078)/2.3026 = 3.
Why is pH defined as a negative logarithm?
pH = −log₁₀[H⁺], where [H⁺] is the hydrogen-ion concentration in moles per liter, typically a tiny fraction such as 10⁻⁷ M for pure water. Taking the negative log10 converts that fraction into a simple positive number (pH 7 for pure water) that's easy to read and compare.
Can I take the negative log of zero or a negative number?
No. The logarithm function is only defined for positive real numbers, so −logb(x) requires x > 0. As x approaches 0 from the positive side, −logb(x) grows without bound; there's no defined value at or below zero, and the base must be positive and not equal to 1.

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