# Negative Log Calculator

URL: https://speedycalculator.com/calculators/math/negative-log/
Last updated: 2026-10-30

> The negative log calculator finds −log_a(b), the negative logarithm used for pH, pOH, pKa and similar "p" scales, in base 10 or any other base.

## Inputs

- Number b
- Base a

## Results

- −log_a(b)
- log_a(b)

## Formula

```
−log_a(b) = −ln b / ln a  (pH = −log₁₀[H⁺])
```

## Explanation

Chemists use negative logarithms to turn tiny concentrations into convenient numbers: a hydrogen ion concentration of 0.0000001 mol/L becomes pH 7. The same trick gives pKa, pOH and pKw.

### How to use the Negative Log Calculator

- **Number b**
- **Base a**

Results update as you type, and everything is calculated in your browser.

### How it works

Enter the number b and the base a (10 for pH-type scales).

### Inputs explained

FieldWhat to enterNumber bA numberBase aA number

### What each result means

- **−log_a(b)** = -ln([Number b])÷ln([Base a])
- **log_a(b)** = ln([Number b])÷ln([Base a])

### Example values

How the main result changes when **Number b** changes and everything else stays at the example values above:

Number b−log_a(b)5.0000e-87.3017.5000e-87.12491.0000e-771.2500e-76.90311.5000e-76.82392.0000e-76.699

## Worked example

Inputs — Number b: 1e-07; Base a: 10.

Results — **−log_a(b) = 7**; log_a(b) = -7.

## Key terms

- **pH**: −log₁₀ of the hydrogen ion concentration.

## FAQ

**What is the pH of 10⁻⁷ mol/L H⁺?**
7.

**Why use a negative log?**
Concentrations are usually below 1, so their logs are negative; the minus sign makes the scale positive.

**What is the pH of 0.01 M HCl?**
2.

**Does a lower pH mean more acid?**
Yes – each unit is ten times more H⁺.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)