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
| Field | What to enter |
|---|---|
| Number x | A number |
| Original base a | A number |
| New base b | A 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 x | log_a(x) |
|---|---|
| 25 | 4.6439 |
| 37.5 | 5.2288 |
| 50 | 5.6439 |
| 62.5 | 5.9658 |
| 75 | 6.2288 |
| 100 | 6.6439 |
Formula
log_a(x) = log_b(x) / log_b(a)
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.
Frequently asked questions
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.