MultiConvers

Log Calculator

Ask “what power of this base gives my number?” in base 10, e, 2, or a custom base — pH, decibels-adjacent homework, and binary length, not a log of a negative.

How to use

  1. Pick common log (10), natural log (e), binary log (2), or a custom base.
  2. Enter a positive number. Zero and negatives are rejected.
  3. Read the exponent. log₁₀(1000) is 3; log₂(8) is 3; ln(e) is 1.
  4. Check with the exponent calculator: base^result should return the number.

How it works

A logarithm is an exponent. log₁₀(1000) = 3 because 10³ = 1000. ln(e) = 1 because e¹ = e. log₂(8) = 3 because 2³ = 8. Custom bases use the change-of-base formula ln(x)/ln(b).

The argument must be positive. There is no real log of 0 or of −5. Base 1 is undefined (1 to any power is 1, never 7). Bases must be positive and not 1.

Common log shows up in pH and in “orders of magnitude.” A jump from 10^2 to 10^5 is 3 common-log units. Natural log shows up in continuous growth. Binary log is “how many bits”: log₂(256) = 8.

This is not the scientific calculator’s full expression parser, and it is not a decibel tool (those mix a 10 or 20 factor with a ratio). Inverse work is exponentiation, not another log.

log₁₀(1) = 0 for any valid base: the power that returns 1 is 0. log_b(b) = 1. Those two identities are the usual sanity check after a custom-base entry. A pH of 3 is not log₁₀(3); it is −log₁₀ of the hydrogen-ion activity, which this page does not compute.

Formula

log_b(x) = ln(x) / ln(b)
log₁₀(1000) = 3
ln(e) = 1
log₂(8) = 3

Worked examples

  • Common log

    log₁₀(1000) → 3

  • Binary log

    log₂(8) → 3

Common logs that should come out clean

ExpressionValue
log₁₀(1)0
log₁₀(10)1
log₁₀(1000)3
ln(e)1
log₂(8)3
log₂(256)8

Useful notes

  • log on the scientific calculator is also base 10; ln is natural.
  • Argument > 0; base > 0 and ≠ 1.
  • Undo with the exponent calculator.

FAQ

What is the difference between log and ln?

On this page, common log is base 10 and ln is base e ≈ 2.71828. Both are logarithms; only the base changes.

How do I change the base of a logarithm?

Divide natural logs: log_b(x) = ln(x) / ln(b). That is what the custom-base mode does.

Why is log(0) rejected?

There is no real power of a positive base that equals 0. The real logarithm is only defined for positive numbers.

What is log₂(256)?

8, because 2^8 = 256.

How do I get log base 5 of 125?

Choose custom base 5 and enter 125, or compute ln(125)/ln(5). The result is 3.

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