Logarithm Calculator
Undoing an Exponent
A logarithm answers a specific question: what power do I need to raise this base to, in order to get this number? It's the mirror image of exponentiation, and three variants cover almost everything encountered in practice — natural log (base e), common log (base 10), and a fully custom base for anything else. This calculator supports all three, enforcing the domain rule that logarithms only exist for positive numbers.
The Formulas
Common Log: log₁₀(x)
Custom Base: log_b(x) = ln(x) / ln(b)
Every custom-base logarithm is computed via the change-of-base identity, dividing the natural log of the number by the natural log of the base. The base itself must be positive and not equal to 1, since log base 1 is undefined (1 raised to any power is always 1).
Where Logarithms Appear
- Sound and signal measurement — decibels are a base-10 logarithmic scale.
- Earthquake magnitude — the Richter scale is logarithmic, so each whole step represents a tenfold increase in amplitude.
- pH chemistry — pH is the negative base-10 logarithm of hydrogen ion concentration.
- Finance and computer science — continuous compounding uses natural logs; algorithmic complexity analysis often involves base-2 logs.
| Expression | Result |
|---|---|
| ln(10) | 2.302585 |
| log₁₀(1000) | 3 |
| log₂(8) | 3 |
How to Use This Calculator
- Choose a mode: Natural Log, Common Log, or Custom Base Log.
- Enter the Number (x) you want the logarithm of.
- For Custom Base, also enter the Base.
- Select Calculate to see the result.
Related Calculations
Need to reverse a logarithm back into its original value? The Inverse Logarithm Calculator handles antilogs directly, and the Exponent Calculator covers raising any base to any power.