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Common and natural logarithm calculator

Calculate the common logarithm (base 10) and the natural logarithm (base e) of any positive number at once.

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Common logarithm (log₁₀)

2

Natural logarithm (ln)

4.605170186

The logarithm is the exponent the base must be raised to in order to get the number:

  • log₁₀ x = y ⇔ 10^y = x (log₁₀ 1000 = 3)
  • ln x = y ⇔ e^y = x, with e ≈ 2.71828 (ln e = 1)
  • How they relate: ln x = log₁₀ x × ln 10 ≈ log₁₀ x × 2.302585

Use cases

  • Solving exponential equations in homework.
  • Working out pH, decibels or other logarithmic scales.
  • Getting the doubling time under continuous growth (ln 2 ÷ rate).

How it works

  1. 1

    Type the number

    It must be greater than 0. Decimals are accepted.

  2. 2

    Read both results

    Common logarithm on top, natural logarithm below. Both can be copied.

Technical details

  • Only defined for numbers greater than 0: with 0 or negatives the calculator shows "-".
  • Between 0 and 1 the logarithm is negative (log₁₀ 0.01 = −2); at 1 it's 0 in any base.
  • For another base b, divide: log_b x = ln x ÷ ln b (e.g. log₂ 8 = ln 8 ÷ ln 2 = 3).
  • Result with up to 10 significant digits.

Frequently asked questions

What's the difference between log and ln?

Only the base: "log" usually means base 10 and "ln" base e. In some calculators and programming languages a bare "log" is the natural log, so it's worth checking.

Why is there no logarithm of 0?

Because no power of a positive number equals 0: the more negative the exponent, the closer it gets to 0 without ever reaching it.

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