Negative Log Calculator (-log)
Enter a positive number and choose a base to compute its negative logarithm instantly. The most common use is pH, where pH = -log10([H+]), but the same operation appears in decibel formulas, Shannon entropy, and many other scientific fields. This calculator supports base 10 (common log), base e (natural log), base 2 (binary/Shannon), and any custom positive base you need. Results update as you type, with a full worked example in the steps panel.
Formula
Worked example
For x = 0.001 and base 10: -log10(0.001) = -(log10(0.001)) = -(log10(10^-3)) = -(-3) = 3. This is why a hydrogen-ion concentration of 0.001 mol/L gives pH 3.
What is the negative logarithm?
The negative logarithm of a number x in base b is written -log_b(x) and equals the ordinary logarithm multiplied by -1. It answers the question: "To what power must I raise b so that b^n = x, and what is the opposite of that power?" The most famous use is pH, where pH = -log10([H+]) and [H+] is the molar concentration of hydrogen ions. Because concentrations in chemistry are often tiny fractions like 0.001 mol/L, the negative log converts them into positive, easy-to-work-with numbers like 3. The same pattern appears in decibels, information entropy, and natural-decay equations.
The reciprocal identity and key properties
The single most useful identity is -log_b(x) = log_b(1/x). This means you can always rewrite a negative log as the ordinary log of the reciprocal, and many textbook problems are written that way. Other handy properties: -log_b(1) = 0 for any base; -log_b(x*y) = -log_b(x) - log_b(y); and -log_b(x^n) = -n * log_b(x). When x is between 0 and 1 the result is positive; when x equals 1 the result is 0; when x is greater than 1 the result is negative. These sign rules are the source of most beginner confusion.
Base choices and their real-world uses
Base 10 (common log) is used for pH, the Richter earthquake scale, and sound intensity in decibels. Base e (natural log, ln) appears in thermodynamics, Gibbs free energy, radioactive decay, and continuous compounding. Base 2 (binary log) gives the information content of events in bits, which is the foundation of Shannon entropy: H = -sum p(x) * log2(p(x)). The change-of-base rule means any base can be computed from any other: -log_b(x) = -ln(x)/ln(b).
Negative log versus log of a negative number
These are two completely different things. The negative log, -log_b(x), applies to a positive x and simply negates the result. The logarithm of a negative number, log_b(-x), is undefined in real arithmetic because no real power of a positive base produces a negative number. If your input x is negative or zero this calculator returns no result because the operation is not defined over the real numbers.
Common negative log values (base 10)
| x (input) | -log10(x) | pH interpretation | Category |
|---|---|---|---|
| 0.000001 | 6 | Very acidic | Strong acid |
| 0.00001 | 5 | Acidic | Moderate acid |
| 0.0001 | 4 | Acidic | Weak acid |
| 0.001 | 3 | Acidic | Typical acid |
| 0.01 | 2 | Acidic | Stomach acid range |
| 0.1 | 1 | Mildly acidic | Lemon juice range |
| 1 | 0 | Neutral boundary | Pure acid |
| 10 | -1 | Negative (x > 1) | Above 1 mol/L |
Key reference points. pH = -log10([H+]) for aqueous solutions at 25 C.
Frequently asked questions
What is -log(0.001) in base 10?
-log10(0.001) = 3. Since 0.001 = 10^(-3), the logarithm is -3 and its negative is 3. In chemistry this corresponds to a pH of 3, which is acidic.
Is -log(x) the same as log(1/x)?
Yes. This is the reciprocal identity: -log_b(x) = log_b(1/x). The reason is that log_b(x^n) = n * log_b(x), so log_b(1/x) = log_b(x^-1) = -1 * log_b(x) = -log_b(x). You can freely switch between the two forms.
Why is the negative log positive when x is less than 1?
When 0 < x < 1, the ordinary logarithm log_b(x) is negative (because you need a negative exponent to get a fraction). Multiplying a negative number by -1 gives a positive result, so -log_b(x) is positive for any x in (0, 1).
How do I calculate pH from hydrogen-ion concentration?
Use pH = -log10([H+]), where [H+] is the molar concentration in mol/L. For [H+] = 0.001 mol/L: pH = -log10(0.001) = -(-3) = 3. Enter 0.001 as x and select base 10 in this calculator to confirm.
Can the base of a logarithm be negative?
No, not in standard real-number arithmetic. The base must be a positive number other than 1. A base of 1 is excluded because 1 raised to any power is always 1, making it impossible to express other numbers. Negative bases lead to complex (imaginary) numbers for most exponents.
What does -log2(p) mean in information theory?
-log2(p) is the self-information or surprisal of an event with probability p, measured in bits. A fair coin flip (p = 0.5) has -log2(0.5) = 1 bit of information. A rare event (p = 0.001) has about 10 bits because it is harder to predict. Shannon entropy sums -p * log2(p) over all outcomes.