Log Calculator
Enter a number and a base below to instantly calculate the logarithm.
What Is a Logarithm?
A logarithm is the inverse of an exponent. If b^y = x, then log base b of x equals y. Logarithms turn multiplication problems into addition problems, which is why they were used for calculation long before calculators existed, and why they still show up in anything that grows or decays exponentially.
Worked example
Find log base 2 of 32.
Ask: 2 raised to what power gives 32?
2^5 = 32, so log₂(32) = 5
A second example
Find log base 5 of 125.
Ask: 5 raised to what power gives 125?
5^3 = 125, so log₅(125) = 3
Keep going
- Logarithms and exponents undo each other by definition; the exponent calculator runs that same relationship in the opposite direction. Exponent Calculator
Frequently Asked Questions
What is a logarithm?
A logarithm answers the question: what power do you need to raise the base to, to get this number? log base 10 of 100 is 2, because 10² = 100.
What's the difference between log and ln?
"log" (without a written base) usually means base 10. "ln" means natural log, which uses e (approximately 2.71828) as its base instead. Both are logarithms, just with different bases.
Why can't you take the log of a negative number or zero?
No power of a positive base ever produces a negative number or zero, so there's no exponent that solves the equation. The logarithm is undefined for inputs less than or equal to 0.
How are logarithms and exponents inverse operations?
Applying one undoes the other when the base matches: log_b(b^x) = x, and b^(log_b x) = x. That's exactly why a logarithm answers "what exponent gives this result" — it's built to reverse exponentiation.
How do you calculate a log in a base your calculator doesn't have a button for?
Use the change-of-base formula: log_b(x) = log(x) / log(b), using any base you do have (base 10 or natural log both work). For example, log₃(50) = log(50) / log(3).
What's a common log mistake?
Assuming log(a + b) equals log(a) + log(b). It doesn't — that identity is false. The real rules are log(a × b) = log(a) + log(b) and log(a / b) = log(a) − log(b), for multiplication and division, not addition.
Related Calculators
- Quadratic Formula Calculator — Algebra
- Exponent Calculator — Algebra
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- System of Equations Calculator — Algebra