Instantly compute logₐ(x) from a value and base. Also shows natural log (ln), common log (log₁₀) and binary log (log₂), with a function graph.
The logarithm logₐ(x) answers "to what power must the base a be raised to get x?" Change of base uses the ratio of natural logs.
The logarithm logₐ(x) is the exponent that answers "to what power must the base a be raised to get x?" It is the inverse of exponentiation — for example log₂(8) = 3 because 2³ = 8. Logs turn multiplication into addition and help handle quantities that span many orders of magnitude.
Enter the positive number you want the log of in "Value" and the positive base (other than 1) in "Base". Alongside the result, the calculator also shows the natural log (ln), common log (log₁₀) and binary log (log₂).
Change of base: logₐ(x) = ln(x) / ln(a). ln(x) = logₑ(x) is the natural log and log(x) = log₁₀(x) is the common log. By definition, logₐ(x) = y ⇔ aʸ = x.
The result tells you what power the base a must be raised to in order to reach the value x. Each time it increases by 1, the value multiplies by the base. If the value is ≤ 0, or the base is ≤ 0 or equal to 1, the logarithm is undefined and you get an error.
ln is the natural log with base e (≈2.718); log usually means the common log with base 10. This calculator shows both at once.
It depends on the use case: base 2 in information theory and computing, base 10 in many sciences, and base e (natural log) in calculus and analysis.
No. The real logarithm is only defined for a positive value (x > 0), so entering 0 or a negative number returns an error.
logₐ(x) = ln(x) / ln(a). As long as you can compute a natural (or common) log, you can find the log to any base.
They are used for pH, decibels, earthquake magnitude, compound-interest time calculations and many other quantities that vary over a wide range.
This calculator is a general-purpose tool for learning and quick checks.