Log Calculator
Calculate the logarithm of a number at any base.
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About this calculator
What Is a Logarithm?
A logarithm answers the question "what power do I raise this base to in order to get this number?" It is the inverse of exponentiation: if bʸ = x, then logb(x) = y. For example, log₁₀(1000) = 3 because 10³ = 1000, and log₂(8) = 3 because 2³ = 8. Enter any two of the three values — base, number or result — in this log calculator and it solves for the third. It also accepts "e" as a base for natural logarithms.
Common Bases
- Base 10 (log): the common logarithm, used in science and engineering — pH, decibels and the Richter scale are all logarithmic.
- Base e (ln): the natural logarithm, with e ≈ 2.71828; it dominates calculus, physics and continuous growth models.
- Base 2 (log₂): the binary logarithm, central to computer science — it tells you how many bits you need or how many times you can halve something.
Worked Example
log₁₀(1000) = 3, since 10³ = 1000. The natural log of 1000 is 6.9078.
The Basic Log Rules
- Product: logb(x × y) = logbx + logby
- Quotient: logb(x ÷ y) = logbx − logby
- Power: logb(xʸ) = y × logbx
- Change of base: logb(x) = logk(x) ÷ logk(b) — this is how calculators evaluate any base using only log or ln.
- Special values: logb(1) = 0, logb(b) = 1, ln(eˣ) = x.
The logarithm of zero or of a negative number is not defined for real numbers, and the base must be positive and not equal to 1.
Why Logarithms Are Useful
Logarithms turn multiplication into addition, which is why slide rules and log tables were once an engineer's main tools. Today they let us handle quantities that span many orders of magnitude, solve for the exponent in growth and decay problems (for example, how long until an investment doubles), and compress huge ranges onto readable scales. Pair this tool with the Exponent Calculator, since each undoes the other, and the Half-Life Calculator for decay problems.
Worked example
The log base 10 of 1,000 is 3, since 10³ = 1,000.
Frequently asked questions
What does a logarithm actually mean?
log_b(x) answers "what power do I raise b to, to get x?" It's the inverse operation of exponentiation, the same way division is the inverse of multiplication.
What's the difference between log, ln, and log10?
They're the same operation at different bases: "log10" (or often just "log") uses base 10, "ln" (natural log) uses base e ≈ 2.71828. This calculator computes any base you enter, plus ln and log10 for reference.
Why can't I take the log of a negative number or zero?
No real power of a positive base produces a negative result or zero, so the logarithm is undefined there — this is why the input must be a positive number.
