Logarithm Calculator
A logarithm answers one question: what power do I have to raise this base to, in order to get this number?
What a logarithm is
A logarithm is the inverse of exponentiation. These two statements say exactly the same thing:
Because 10³ = 1000, log₁₀(1000) = 3. The logarithm is simply the exponent, extracted and named.
Three bases dominate in practice:
- Base 10 (common log, written log) — used wherever orders of magnitude matter: decibels, pH, the Richter scale.
- Base e (natural log, written ln), where e ≈ 2.71828 — the base that appears naturally in calculus, continuous growth and decay, and compound interest.
- Base 2 (binary log) — computer science, information theory, and algorithm complexity.
The log laws
The first law is the historically important one. It converts multiplication into addition, which is why logarithms were invented by Napier in 1614 and why slide rules and log tables dominated calculation for three and a half centuries. Multiplying two seven-digit numbers by hand is laborious; adding their logarithms is not.
The change of base formula is what lets a calculator with only ln and log₁₀ compute any base at all — and it is exactly what this tool does internally.
Logarithmic scales in the real world
Logarithmic scales appear wherever quantities span many orders of magnitude, or where human perception is itself logarithmic:
| Scale | Measures | Each step means |
|---|---|---|
| Decibel | Sound intensity | +10 dB = 10× the power |
| Richter | Earthquake magnitude | +1 = ~31.6× the energy |
| pH | Acidity | −1 = 10× more H⁺ ions |
| Stellar magnitude | Brightness | −1 = ~2.512× brighter |
| Octave | Musical pitch | +1 = 2× the frequency |
This is why a magnitude 7 earthquake is not slightly worse than a magnitude 6 but roughly thirty times more energetic, and why halving the power of a speaker barely changes the perceived loudness.
Frequently asked questions
Why can't you take the log of a negative number?
Because no real power of a positive base produces a negative result. There is no x for which 10ˣ = −5. Complex analysis extends logarithms to negative and complex numbers, but the answer is then complex and multivalued.
What is e and why does it matter?
e ≈ 2.71828 is the limit of (1 + 1/n)ⁿ as n grows without bound — the value continuous compounding converges to. Its defining property is that the function eˣ is its own derivative, which makes it the natural base for anything involving continuous growth or decay.
What is an antilog?
The inverse operation: raising the base to the power of the logarithm. If log₁₀(x) = 2.5 then the antilog is 10^2.5 ≈ 316.2. On calculators it is usually the 10ˣ or eˣ key, often the shifted function of the log and ln keys.