Logarithm rules
| Rule | Formula |
|---|
| Product | log_b(xy) = log_b(x) + log_b(y) |
| Quotient | log_b(x/y) = log_b(x) − log_b(y) |
| Power | log_b(xⁿ) = n · log_b(x) |
| Change of base | log_b(x) = ln(x) ÷ ln(b) |
| Special values | log_b(1) = 0, log_b(b) = 1 |
Where logarithms show up
- pH: pH = −log₁₀[H⁺]. Each step on the pH scale is a tenfold change in acidity.
- Decibels: sound level uses 10 × log₁₀ of an intensity ratio, so +10 dB is ten times the intensity.
- Earthquakes: each whole step up the magnitude scale releases about 32 times more energy.
- Doubling time: money at 7% a year doubles in ln(2) ÷ ln(1.07) ≈ 10.2 years.
- Computer science: binary search through a million items takes about log₂(1,000,000) ≈ 20 steps.
Solving for an unknown exponent? The exponent calculator can do that directly. For log, ln and trig in one place, use the scientific calculator.
Last reviewed October 2026.
Frequently asked questions
What is a logarithm?
A logarithm answers the question "what power do I raise the base to, to get this number?". log₂(8) = 3 because 2³ = 8. Logarithms are the inverse of exponents.
What is the difference between log and ln?
log usually means base 10 (the common logarithm), so log(100) = 2. ln means base e ≈ 2.71828 (the natural logarithm), used in calculus, growth and decay. In some programming languages and advanced math texts, log means ln, so check the context.
How do I calculate a log with any base?
Use the change of base formula: log_b(x) = ln(x) ÷ ln(b), or log(x) ÷ log(b). For example, log₂(32) = ln(32) ÷ ln(2) = 5. The calculator does this for you.
Why can't I take the log of a negative number or zero?
A positive base raised to any real power is always positive, so no real power can give zero or a negative number. The log of 0 or a negative number is undefined among the real numbers.
What is an antilog?
The antilog undoes a logarithm. The antilog of y in base b is b^y. If log(x) = 3, the antilog gives x = 10³ = 1000.