Logarithm Calculator

log_b(x) for any base b > 0, b ≠ 1, x > 0. Common bases shown side-by-side.

Inputs

Result

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General calculation reads

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How to use this calculator

  • Enter x and the base.
  • Read the log; the breakdown shows base-10, natural, and base-2 logs for reference.

About this calculator

A logarithm answers: "to what power must I raise the base to get x?" log₁₀(100) = 2 because 10² = 100. Common bases: base 10 (log), base e ≈ 2.718 (ln, natural log), base 2 (used in computer science). Change-of-base formula lets you compute log in any base from natural log: log_b(x) = ln(x) / ln(b).

How it works — the formula

logₐ(x) = y ⟺ aʸ = x (general) log(x) := log₁₀(x) (common log) ln(x) := logₑ(x), e ≈ 2.71828 (natural log) Change of base: logₐ(x) = ln(x) / ln(a)

The logarithm logₐ(x) is the exponent y to which the base a must be raised to produce x. The base must be positive and not 1; the argument must be strictly positive. The common log uses base 10 (standard for orders of magnitude — pH, decibels, Richter scale), and the natural log uses base e (standard for continuous-rate models — radioactive decay, compound interest, population growth). The change-of-base identity lets any calculator with one log function compute any other.

Sources: NIST DLMF §4 — Elementary functions: logarithm · NIST DLMF §4.6 — Logarithm series and identities · Wolfram MathWorld — Logarithm

Worked examples

Example 1
Common log
Inputs:
log₁₀(1000)
Output:
3 (because 10³ = 1000)
Example 2
Natural log
Inputs:
ln(e²)
Output:
2 (because eˣ and ln are inverses)
Example 3
Change of base
Inputs:
log₂(10) via natural log
Output:
log₂(10) = ln(10) / ln(2) ≈ 3.3219

Limitations

  • log of zero is −∞ as a limit; log of a negative number is complex (not real).
  • Floating-point representation has ~15–16 significant digits; logarithms of very large or very small numbers may lose precision in the trailing digits.
  • Different mathematical communities use "log" to mean either log₁₀ (most science/engineering) or logₑ (most pure math) — clarify the base when ambiguous.
  • The series ln(1 + x) = x − x²/2 + x³/3 − … converges only for −1 < x ≤ 1; for larger arguments, use the change-of-base identity or built-in implementations.

Computed with the platform's IEEE 754 Math.log family; agreement with NIST tabulated values exceeds 12 significant digits across the typical input range.

Frequently asked

What is the natural log?+
log base e ≈ 2.71828, written ln(x). Appears throughout calculus and physics because the derivative of ln(x) is 1/x.
Can log of a negative number be computed?+
Not over the real numbers. Complex logs exist but are multi-valued; this calculator returns "—" for non-positive x.
Why are logs useful?+
They turn multiplication into addition: log(ab) = log(a) + log(b). This is why slide rules worked, and why log scales (decibels, pH, Richter) compress huge ranges into readable numbers.
What is log(1)?+
0 in any base — because b⁰ = 1 for any positive base b.
What is log(0)?+
Undefined — approaches negative infinity as x → 0⁺. The calculator rejects x ≤ 0.

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