Log Calculator – ln, log₁₀ & Any Base with Steps

Logarithms invert exponentiation: log₁₀(1000) = 3 because 10³ = 1000, and ln(e²) = 2 because the natural base e satisfies e² ≈ 7.389. The calculator covers ln (base e ≈ 2.71828), common log (base 10) and any custom base through the identity log_b(x) = ln(x)/ln(b) — for instance log₂(64) = ln 64/ln 2 = 4.159/0.693 = 6. The three working rules are demonstrated with numbers: product rule log(2×50) = log 2 + log 50 = log 100 = 2; quotient rule log(100/10) = 2 − 1 = 1; power rule log(2⁵) = 5×log 2 ≈ 1.505. Each result is paired with its exponential check (2⁶ = 64) so you can verify answers in both directions, and inputs are validated: logarithms accept only positive arguments, so log(0) and log(−5) are rejected with an explanation rather than a cryptic error.

1. 1

   Pick the base

   e for natural, 10 for common, or type any base like 2

2. 2

   Enter a positive argument

   The value x whose logarithm you need

3. 3

   Compute via ln

   The change-of-base division is displayed for custom bases

4. 4

   Check the inverse

   The equivalent b^result is printed to confirm

• 1

  ln e = 1 and log₁₀ 10 = 1 — bases log to 1 on themselves

• 2

  log(xʳ) = r·log x turns powers into multiplication

• 3

  The argument must be positive; log(0) is undefined

• Computing log(x+y) as log x + log y — only products split

• Mixing bases: log₁₀(e) ≈ 0.434 is not 1

• Believing log 1 equals 0 is right, but log 0 is not 1 — it is undefined

• Dropping the argument positivity rule and typing 0 or negatives