Logarithm Calculator
log₁₀(x)
Understanding Logarithms
A logarithm is the inverse of exponentiation. If bʸ = x, then log_b(x) = y. In words: the logarithm base b of x is the exponent you raise b to in order to get x. Logarithms were invented to simplify calculations before electronic calculators — they convert multiplication into addition and powers into multiplication, which is why slide rules (based on logarithmic scales) were essential tools for engineers and scientists for centuries.
The three most common logarithms are base 10 (common logarithm, written log or log₁₀), base e (natural logarithm, written ln, where e ≈ 2.71828), and base 2 (binary logarithm, common in computer science). log₁₀(1000) = 3 because 10³ = 1000. ln(e) = 1 because e¹ = e. log₂(8) = 3 because 2³ = 8. Each base serves different fields: base 10 for scientific notation and pH scales, base e for calculus and continuous growth models, base 2 for information theory and algorithm analysis.
The change-of-base formula lets you compute any logarithm using natural or common logs: log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b). This is how calculators evaluate custom-base logarithms internally. For example, log₃(81) = ln(81) / ln(3) = 4.0814 / 1.0986 ≈ 4, which is correct since 3⁴ = 81.
Logarithms have important properties that mirror exponent rules. log(ab) = log(a) + log(b) — the log of a product equals the sum of logs. log(a/b) = log(a) − log(b). log(aⁿ) = n × log(a). These properties are the foundation of logarithmic scales like the Richter scale for earthquakes, decibels for sound intensity, and the pH scale for acidity — all of which compress enormous ranges into manageable numbers.
Important restrictions: the input to a logarithm must be positive (x > 0). log(0) and log(negative numbers) are undefined in the real number system. The base must be positive and not equal to 1. In applications, logarithms model exponential decay (radioactive half-life), solve equations like 2ˣ = 100 (x = log₂(100)), and appear in the formula for compound interest with continuous compounding. Select the log type, enter your value, and get an instant result.
Examples
| Example | Result |
|---|---|
| log₁₀(1000) | 3 |
| log₁₀(100) | 2 |
| ln(1) | 0 |
| log₂(8) | 3 |
| log₁₀(0.01) | −2 |
| log₂(32) | 5 |
| log₃(81) | 4 |
Frequently asked questions
The natural logarithm uses base e (≈ 2.71828). ln(x) is the exponent that e must be raised to in order to get x. ln(e) = 1.
Not in the real number system. Logarithms are only defined for positive inputs. log(−5) is undefined for real numbers.
Use the custom base option with base 2, or apply the change-of-base formula: log₂(x) = ln(x) / ln(2). Example: log₂(8) = 3.