Logarithm Calculator
Logarithm of a number in base 10, e, 2 or any custom base.
Quick answer
- Formula: log_b(x) = ln(x) ÷ ln(b) — the power to which b must be raised to give x
- Worked example — Logarithm in the selected base: 2.0000 (full example below).
Logarithm Calculator
| Logarithm in the selected base | 2.0000 |
| ln (base e) | 4.6052 |
| log₁₀ | 2.0000 |
| log₂ | 6.6439 |
log₁₀(100) = 2, because 10² = 100. The same number in base 2 is about 6.644, and its natural log is about 4.605.
A logarithm answers the question: to what power must I raise this base to get that number? log₁₀(1000) = 3 because 10³ = 1,000. Logarithms turn multiplication into addition, which is why they appear everywhere growth is measured.
Enter a positive number and pick a base — 10 for common logs, e for the natural log (ln), 2 for binary, or any custom base. All four common versions are shown together so you can compare them.
The formula
Worked example
log₁₀(100) = 2, because 10² = 100. The same number in base 2 is about 6.644, and its natural log is about 4.605.
The example is pre-loaded in the result panel — press Calculate with your own numbers to replace it.
Frequently asked questions
Why is there no logarithm of zero or a negative number?
Because no power of a positive base ever reaches zero or goes negative — 10^x is always positive. The logarithm is therefore undefined (in real numbers) for values of zero or below.
Where are logarithms used in real life?
Earthquake magnitude (Richter), sound volume (decibels), acidity (pH), and the number of steps in binary search all use logarithmic scales, because the underlying quantities span enormous ranges.
How are logs and exponents related?
They are inverses: log_b(x) = y exactly when b^y = x. If you know 2^10 = 1,024, then log₂(1024) = 10.
Category: Math calculators
Sources & methodology
This calculator applies the formula shown on this page exactly as written, using the standard definitions for its inputs. Results are computed from the values you enter — no data is stored — and are rounded only for display.
Content last reviewed by CalcHub Editorial.
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