# Change of Base Calculator

URL: https://speedycalculator.com/calculators/math/change-of-base/
Last updated: 2026-10-30

> The change of base calculator evaluates log_a(x) using logarithms in another base b, showing the formula log_a x = log_b x / log_b a step by step.

## Inputs

- Number x
- Original base a
- New base b

## Results

- log_a(x)
- log_b(x)
- log_b(a)

## Formula

```
log_a(x) = log_b(x) / log_b(a)
```

## Explanation

Most calculators only have ln and log₁₀ keys. The change of base formula lets you compute a logarithm in any base from those – for example log₂ 50 = ln 50 / ln 2.

### How to use the Change of Base Calculator

- **Number x**
- **Original base a**
- **New base b**

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

### How it works

Enter the number x, its original base a and the new base b you want to work in (10 or e are common).

### Inputs explained

FieldWhat to enterNumber xA numberOriginal base aA numberNew base bA number

### What each result means

- **log_a(x)** = ln([Number x])÷ln([Original base a])
- **log_b(x)** = ln([Number x])÷ln([New base b])
- **log_b(a)** = ln([Original base a])÷ln([New base b])

### Example values

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

Number xlog_a(x)254.643937.55.2288505.643962.55.9658756.22881006.6439

## Worked example

Inputs — Number x: 50; Original base a: 2; New base b: 10.

Results — **log_a(x) = 5.643856189775**; log_b(x) = 1.698970004336; log_b(a) = 0.301029995664.

## Key terms

- **Base**: The number raised to a power in a logarithm.

## FAQ

**Why does the change of base work?**
If y = log_a x then aʸ = x; take log_b of both sides: y·log_b a = log_b x.

**Does the new base matter?**
No – any valid base gives the same answer.

**What is log_a b × log_b a?**
1.

**Can the base be 1?**
No – 1 to any power is 1.


## Sources

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