Logarithms explained
A logarithm answers the question what power gives this number. It is the inverse of an exponent. This guide explains common and natural logs, the logarithm rules, and why logs matter for real-world scales.
What a logarithm is
If 10 to the 3rd is 1000, then the log base 10 of 1000 is 3. A logarithm takes a number and returns the exponent that produced it from a given base. It is exactly the inverse of raising that base to a power.
Common and natural logs
Two bases are used most. The common log, written log, uses base 10 and suits powers of ten. The natural log, written ln, uses the base e, about 2.718, and appears throughout calculus and growth and decay problems. Any base is possible, but these two dominate.
The logarithm rules
| Rule | Formula |
|---|---|
| Product | log(xy) = log x + log y |
| Quotient | log(x/y) = log x − log y |
| Power | log(x^n) = n × log x |
| Log of 1 | log 1 = 0 (any base) |
| Log of the base | log of b to base b = 1 |
Why logs are useful
Logarithms turn multiplication into addition and powers into multiplication, which is why they once powered slide rules and log tables. Today they measure quantities that span huge ranges, such as the decibel scale for sound, the Richter scale for earthquakes, and pH for acidity, where each step is a tenfold change.
- A logarithm is the inverse of an exponent: it returns the power.
- log uses base 10; ln uses base e (about 2.718).
- log(xy) = log x + log y turns products into sums.
- log(x^n) = n log x turns powers into products.
- Logs describe scales like decibels, Richter, and pH.
Related guides
See the Exponents Guide, since logarithms reverse exponents.
FAQ
What is a logarithm?
The exponent a base must be raised to in order to produce a given number. Log base 10 of 1000 is 3 because 10³ = 1000.
What is the difference between log and ln?
log usually means base 10; ln is the natural log, base e (about 2.718).
What is a logarithm used for?
For inverting exponents and for scales that span large ranges, such as sound (decibels) and earthquakes (Richter).