Logarithm Calculator
Compute logarithms in any base, natural log (ln), log base 10, and antilogarithm.
Inputs
Common bases: 10 (common log), 2 (binary), e ≈ 2.71828 (natural).
Result
3.0000000000
ln(x)
6.90775528
Intermediate value used in change-of-base.
ln(base)
2.30258509
Step by step
Change-of-base formula
log_10(1000) = ln(1000) / ln(10)
ln(1000)
= 6.90775528
ln(10)
= 2.30258509
Divide
6.90775528 ÷ 2.30258509
= 3
Note
This is an exact integer: 10^3 = 1000
How it works
The logarithm log_b(x) answers 'b to what power equals x?'. The natural logarithm ln(x) uses base e ≈ 2.71828. The common logarithm log₁₀(x) is standard in engineering. The binary logarithm log₂(x) is central to computer science. The antilogarithm (antilog) is the inverse: it raises the base to the given power, 'undoing' the logarithm.
Formulas
Definition
log_b(x) = y if and only if b^y = x
- b
- Base
- x
- Argument (must be positive)
- y
- Result (the logarithm)
Change of base
log_b(x) = ln(x) / ln(b)
Antilogarithm
antilog_b(x) = b^x
Frequently Asked Questions
What is log base 2 of 8?
3. Because 2³ = 8. In general, log₂(2ⁿ) = n.
What is the difference between log and ln?
In science and engineering, 'log' without a base usually means log base 10. In mathematics, it often means the natural logarithm (ln). This calculator makes the base explicit to avoid ambiguity.
Why is log(0) undefined?
Because no real power of a positive base equals zero. As the argument approaches 0 from the right, log_b(x) approaches −∞, so 0 is a boundary where the function diverges.
What is the antilog of 3 in base 10?
10³ = 1000. The antilog undoes the log: log₁₀(1000) = 3 and 10³ = 1000 are two ways of saying the same thing.