Logarithm Calculator
Compute logarithms in any base, plus natural log and antilog.
About Logarithm Calculator
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
Worked example
With the default values below, Logarithm Calculator returns Result of 3.0000000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Operationone of log_b(x) — log of x in base b, ln(x) — natural logarithm, Antilog: b^x
- log_b(x) — log of x in base b
- Base (b)minimum 0
- 10
- x (argument)minimum 0
- 1,000
- Exponent (x)
- 3
Results
- Result
- 3.0000000000
- ln(x)Intermediate value used in change-of-base.
- 6.90775528
- ln(base)
- 2.30258509
Step by step
- 1Change-of-base formula
- 2ln(1000)= 6.90775528
- 3ln(10)= 2.30258509
- 4Divide= 3
- 5NoteThis is an exact integer: 10^3 = 1000
Assumptions and limits
- The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.