🧮 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.