# Expanding Logarithms Calculator

URL: https://speedycalculator.com/calculators/math/expanding-logarithms/
Last updated: 2026-10-30

> The expanding logarithms calculator splits the logarithm of a product, quotient or power into sums, differences and multiples of simpler logarithms, with the numerical value.

## Inputs

- Expression
- a
- b
- Exponent k
- Logarithm base
- Custom base

## Results

- Expanded form
- Original
- Value
- Rule

## Formula

```
log(a·b) = log a + log b,  log(a/b) = log a − log b,  log(aᵏ) = k·log a
```

## Explanation

Expanding logarithms breaks a complicated logarithm into simple pieces. It is used to differentiate products (logarithmic differentiation), to linearise data and to simplify expressions before solving.

### How to use the Expanding Logarithms Calculator

- **Expression**
- **a**
- **b**
- **Exponent k**
- **Logarithm base**
- **Custom base**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Choose the form, enter the numbers and the base.

## Worked example

Inputs — Expression: Product log(a·b); a: 8; b: 4; Logarithm base: 10 (common log).

Results — **Expanded form = log₁₀(8) + log₁₀(4)**; Original = log₁₀(8 · 4); Value = 1.50514997832; Rule = Product rule: log(a·b) = log a + log b.

## Key terms

- **Power rule**: log(aᵏ) = k log a.

## FAQ

**How do I expand log(x²y)?**
2 log x + log y.

**Is log(a + b) = log a + log b?**
No – there is no rule for the log of a sum.

**Why expand logarithms?**
To simplify derivatives, equations and data analysis.

**Do the rules work in every base?**
Yes.


## Sources

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