🧮 Logarithm Calculator
Find log base b of any value — or ln, log₁₀ and log₂ at once — and reverse it with the antilogarithm.
log10(1000) =
3
6.907755279
ln (base e)
3
log₁₀
9.965784285
log₂
Antilogarithm (bˣ)
The logarithm answers "what power gives this value": logb(x) is the exponent y for which bʸ = x. Computed by change of base, logb(x) = ln(x) ÷ ln(b). The antilog reverses it: bˣ. 🔒 In your browser.
How the logarithm calculator works
A logarithm answers "to what power must the base be raised to get this number?": logₐb(x) is the exponent y with bʸ = x. For any base the tool uses the change-of-base formula, logₐb(x) = ln(x) ÷ ln(b), so it works for base 10, base 2, the natural base e, or any base you type. It also shows ln, log₁₀ and log₂ together, and an antilogarithm (bˣ) that reverses the operation.
The value must be positive and the base positive and not equal to 1 — logarithms of zero or negative numbers, or in base 1, are undefined. Natural log (base e ≈ 2.71828) is written ln; "log" with no base usually means base 10 in engineering and base e in pure maths, which is why picking the base explicitly matters.
Frequently asked questions
How do I calculate a logarithm in any base?
Use the change-of-base formula: logₐb(x) = ln(x) ÷ ln(b), or equivalently log(x) ÷ log(b) with any common log. For example log₂(8) = ln(8)/ln(2) = 3. The tool does this for whatever base you enter.
What is the difference between ln and log?
ln is the natural logarithm, base e (≈ 2.718). "log" usually means base 10 (common log) in science and engineering, but base e in some maths contexts. Because it's ambiguous, this tool lets you set the base explicitly and shows ln, log₁₀ and log₂ side by side.
What is an antilogarithm?
The inverse of a logarithm: if log_b(x) = y then the antilog is bʸ = x. So the antilog base 10 of 3 is 10³ = 1000. The tool has a separate bˣ field for this.
Why can't I take the log of a negative number?
Because no real power of a positive base gives a negative (or zero) result — bʸ is always positive for a positive base. Logs of negatives exist only in complex numbers, which this tool doesn't cover.
What is log base 2 used for?
Base-2 logs count doublings, so they appear throughout computing and information theory — bits of information, algorithm complexity (O(log n)), and octaves in music. log₂(1024) = 10, for instance.
What does log_b(1) equal?
Always 0, for any base, because b⁰ = 1. And log_b(b) = 1, because b¹ = b. These two identities are worth remembering.