ใ’ Logarithm Calculator

Calculate logarithm of any number to any base.

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What is Logarithm Calculator?

A logarithm answers a single question: what power do you need to raise a given base to, in order to get a specific number? It's the inverse operation of exponentiation โ€” where powers multiply repeatedly, logarithms "undo" that multiplication and hand you back the exponent. This calculator computes the logarithm of any positive number to any base you specify, or the natural logarithm (base e) if you leave the base blank.

The Logarithm Formula (Change of Base)

When you want the natural log, the calculator simply returns ln(n). For any other base, it uses the change-of-base formula:

log_base(n) = ln(n) รท ln(base)

Worked Example

Find log base 10 of 1000 โ€” in other words, "10 raised to what power gives 1000?"

logโ‚โ‚€(1000) = ln(1000) รท ln(10) = 3

This checks out because 10ยณ = 1000.

Now find the natural log of eยฒ (where e โ‰ˆ 2.71828, the base of natural logarithms):

ln(eยฒ) = 2

This follows directly from the definition โ€” the natural log of e raised to any power simply returns that power.

How to Use This Calculator

  1. Enter the number you want the logarithm of in the Number field.
  2. Enter the base in the Base field โ€” or leave it blank for the natural log (base e).
  3. Click Calculate to see the result.
๐Ÿ’ก Tip: Logarithms only accept positive numbers as input โ€” there's no real power you can raise a positive base to that produces zero or a negative result, so entering 0 or a negative number won't give a valid answer.

Common Bases You'll Encounter

Base 10 (the "common logarithm") is used in fields like the Richter scale for earthquakes and the decibel scale for sound, since both describe quantities that vary across enormous ranges more conveniently on a compressed logarithmic scale. Base e (the "natural logarithm") turns up throughout calculus, compound growth, and probability, because e itself arises naturally from continuous growth processes. Base 2 is common in computer science, since doubling patterns govern memory addressing and algorithm complexity.

Since logarithms and exponents are inverse operations, our Power Calculator is a natural companion for checking your answer by working backward.

📅 Last reviewed: July 2026 · Formulas verified against RBI/SEBI/IT Dept guidelines.