Log Calculator
This free log calculator computes the logarithm of a number in any base — base 10 (common log), base e (natural log), base 2, or any custom base you choose. Logarithms answer the question "what power do I raise the base to in order to get this number?" and appear throughout math, science, finance, and computing. Enter the number and the base, and the calculator returns the logarithm.
How to Use the Log Calculator
- Enter the number (must be positive).
- Enter the base (10, e, 2, or any custom base).
- Calculate — see the logarithm.
What a Logarithm Is
If bˣ = y, then log_b(y) = x. For example, log₁₀(1000) = 3 because 10³ = 1000, and log₂(8) = 3 because 2³ = 8. The logarithm is the inverse of exponentiation.
Common Logarithm Bases
| Notation | Base | Used In |
|---|---|---|
| log | 10 | Science, pH, decibels |
| ln | e ≈ 2.718 | Calculus, growth/decay |
| log₂ | 2 | Computing, information theory |
The Change of Base Formula
To compute a logarithm in any base using only common or natural logs:
log_b(x) = log(x) ÷ log(b) = ln(x) ÷ ln(b)
For example, log₂(10) = ln(10) ÷ ln(2) ≈ 3.32. The calculator applies this automatically for custom bases.
Logarithm Rules
- Product: log(ab) = log(a) + log(b)
- Quotient: log(a/b) = log(a) − log(b)
- Power: log(aⁿ) = n·log(a)
- log_b(1) = 0 and log_b(b) = 1
Frequently Asked Questions
How do you calculate a logarithm?
Find the power the base must be raised to in order to produce the number. log₁₀(1000) = 3 because 10³ = 1000. The calculator handles any base.
What is the difference between log and ln?
"log" usually means base 10, while "ln" is the natural logarithm with base e. Both measure how many times a base multiplies to reach a number.
What is the change of base formula?
log_b(x) = log(x) ÷ log(b), letting you compute a logarithm of any base using common or natural logs.
Can you take the log of a negative number?
No — logarithms are only defined for positive numbers, since a positive base raised to any real power is always positive.
Is this log calculator free?
Yes — it is completely free, requires no signup, and computes logarithms in any base.